diff --git "a/EdE1T4oBgHgl3EQfEgNA/content/tmp_files/load_file.txt" "b/EdE1T4oBgHgl3EQfEgNA/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/EdE1T4oBgHgl3EQfEgNA/content/tmp_files/load_file.txt" @@ -0,0 +1,552 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf,len=551 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='02890v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='DS] 7 Jan 2023 ERGODICITY AND PERIODIC ORBITS OF p-ADIC (1, 2)-RATIONAL DYNAMICAL SYSTEMS WITH TWO FIXED POINTS I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' SATTAROV, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' ALIEV Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' We consider (1, 2)-rational functions given on the field of p-adic numbers Qp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' In general, such a function has four parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' We study the case when such a function has two fixed points and show that when there are two fixed points then (1, 2)-rational function is conjugate to a two-parametric (1, 2)-rational function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Depending on these two parameters we determine type of the fixed points, find Siegel disks and the basin of attraction of the fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Moreover, we classify invariant sets and study ergodicity properties of the function on each invariant set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' We describe 2- and 3-periodic orbits of the p-adic dynamical systems generated by the two-parametric (1, 2)-rational functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Introduction and preliminaries A function is called a (n, m)-rational function if and only if it can be written in the form f(x) = Pn(x) Qm(x), where Pn(x) and Qm(x) are polynomial functions with degree n and m respectively, Qm(x) is non zero polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' In this paper we study dynamical systems generated by a (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='2-)rational function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Our investigations based on methods of [1], [3], [13]-[17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' For motivations of the study see [2], [4]-[6], [10]-[12] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let us give main definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let Q be the field of rational numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' The greatest common divisor of the positive integers n and m is denotes by (n, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Every rational number x ̸= 0 can be represented in the form x = pr n m, where r, n ∈ Z, m is a positive integer, (p, n) = 1, (p, m) = 1 and p is a fixed prime number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' The p-adic norm of x ∈ Q is given by |x|p = � p−r, for x ̸= 0, 0, for x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' It has the following properties: 1) |x|p ≥ 0 and |x|p = 0 if and only if x = 0, 2) |xy|p = |x|p|y|p, 3) the strong triangle inequality |x + y|p ≤ max{|x|p, |y|p}, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='1) if |x|p ̸= |y|p then |x + y|p = max{|x|p, |y|p}, 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' 46S10, 12J12, 11S99, 30D05, 54H20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Rational dynamical systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' fixed point;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' invariant set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Siegel disk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' complex p-adic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' 1 2 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' SATTAROV, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' ALIEV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='2) if |x|p = |y|p then |x + y|p ≤ |x|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' The completion of Q with respect to p-adic norm defines the p-adic field which is denoted by Qp (see [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' For any a ∈ Qp and r > 0 denote Ur(a) = {x ∈ Qp : |x − a|p < r}, Vr(a) = {x ∈ Qp : |x − a|p ≤ r}, Sr(a) = {x ∈ Qp : |x − a|p = r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' A function f : Ur(a) → Qp is said to be analytic if it can be represented by f(x) = ∞ � n=0 fn(x − a)n, fn ∈ Qp, which converges uniformly on the ball Ur(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Now let f : U → U be an analytic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Denote f n(x) = f ◦ · · · ◦ f � �� � n (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' If f(x0) = x0 then x0 is called a fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' The set of all fixed points of f is denoted by Fix(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' A fixed point x0 is called an attractor if there exists a neighborhood U(x0) of x0 such that for all points x ∈ U(x0) it holds lim n→∞ f n(x) = x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' If x0 is an attractor then its basin of attraction is A(x0) = {x ∈ Qp : f n(x) → x0, n → ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' A fixed point x0 is called repeller if there exists a neighborhood U(x0) of x0 such that |f(x) − x0|p > |x − x0|p for x ∈ U(x0), x ̸= x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let x0 be a fixed point of a function f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Put λ = f ′(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' The point x0 is attractive if 0 < |λ|p < 1, indifferent if |λ|p = 1, and repelling if |λ|p > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' The ball Ur(x0) is said to be a Siegel disk if each sphere Sρ(x0), ρ < r is an invariant sphere of f(x), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' if x ∈ Sρ(x0) then all iterated points f n(x) ∈ Sρ(x0) for all n = 1, 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' The union of all Siegel desks with the center at x0 is said to a maximum Siegel disk and is denoted by SI(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let f : A → A and g : B → B be two maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' f and g are said to be topologically conjugate if there exists a homeomorphism h : A → B such that, h◦f = g ◦h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' The homeomorphism h is called a topological conjugacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Mappings which are topologically conjugate are completely equivalent in terms of their dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' In this paper we consider (1, 2)-rational function f : Qp → Qp defined by f(x) = ax + b x2 + cx + d, x ̸= ˆx1,2 = −c ± √ c2 − 4d 2 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='1) where the parameters of the function satisfy the following conditions a ̸= 0, a, b, c, d, � c2 − 4d ∈ Qp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' We study p-adic dynamical systems generated by the rational function (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' The equa- tion f(x) = x for fixed points of the function (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='1) is equivalent to the equation x3 + cx2 + (d − a)x − b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='2) The equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='2) may have three solutions with one of the following relations: ERGODICITY AND PERIODIC ORBITS 3 (i) one solution having multiplicity three;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' (ii) two solutions, one of which has multiplicity two;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' (iii) three distinct solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Since the behavior of dynamical system depends on the set of fixed points, each of the above mentioned case (i)-(iii) has its own character of dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' In [15] the case (i) was considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' In this paper we consider the case (ii), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=', we investigate the behavior of the trajectories of an arbitrary (1, 2)-rational dynamical system in Qp when there are two fixed points for f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' The case (iii) will be considered in a separate paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' In Section 2 under some assumptions we show that four-parametric function (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='1) is conjugate to a two-parametric (1,2)-rational function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' In Section 3 we study the p-adic dynamics generated by the two-parametric function and give Siegel disks, the basin of attractions and classification of all invariant sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' In Section 4 we investigate ergodicity of this dynamical systems on invariant sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' In Section 5 we describe 2- and 3-periodic orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' A function conjugate to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='1) Denote by x1 and x2 the two solutions of the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='2), where x2 has multiplicity two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Then we have x3 + cx2 + (d − a)x − b = (x − x1)(x − x2)2 and \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 x1 + 2x2 = −c x2 2 + 2x1x2 = d − a x1x2 2 = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='1) Let homeomorphism h : Qp → Qp be defined by h(t) = t+x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' We note that, the function f is topologically conjugate to function h−1 ◦ f ◦ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' We have (h−1 ◦ f ◦ h)(t) = −x2t2 + Bt t2 + Dt + B , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='2) where B = x2 2 + cx2 + d and D = 2x2 + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' In [13] the case x2 ̸= 0 is studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Thus in this paper we consider the case x2 = 0 in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' If x2 = 0, then B = d = a and D = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Thus we have the following proposition Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Any (1,2)-rational function having two distinct fixed points is topologically conjugate to one of the following functions f(x) = ax2 + bx x2 + cx + b, ab(a − c) ̸= 0, a, b, c ∈ Qp, and f(x) = ax x2 + cx + a, ac ̸= 0, a, c, ∈ Qp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3) where x ̸= ˆx1,2 = −c± √ c2−4a 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' We study the dynamical system (Qp, f) with f given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' 4 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' SATTAROV, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' ALIEV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' p-Adic dynamics of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3) Note that, the function (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3) has two fixed points x1 = 0 and x2 = −c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' We have f ′(x1) = 1 and f ′(x2) = 1 − c2 a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Thus, the point x1 is an indifferent point for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=', x1 is a center of some Siegel disk SI(x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' In this section we determine the character of the fixed point x2 for each cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Then we find Siegel disk or basin of attraction of the fixed point x2, when x2 is indifferent or attractive, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' In the case where x2 is repelling, we find open ball Ur(x2), such that the inequality |f(x) − x2|p > |x − x2|p holds for all x ∈ Ur(x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Moreover, we study a relation between the sets SI(x1) and SI(x2) when x2 is an indifferent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' For any x ∈ Qp, x ̸= ˆx1,2, by simple calculations we get |f(x)|p = |x|p · |a|p |x − ˆx1|p|x − ˆx2|p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='1) Denote P = {x ∈ Qp : ∃n ∈ N ∪ {0}, f n(x) ∈ {ˆx1, ˆx2}}, α = min{|ˆx1|p, |ˆx2|p} and β = max{|ˆx1|p, |ˆx2|p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='2) Since ˆx1 + ˆx2 = −c, we have |c|p ≤ α for α = β and |c|p = β for α < β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Also, since ˆx1ˆx2 = a, we have |a|p = αβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' The p-adic dynamical system generated by the function (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3) has the following properties: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' SI(x1) = Uα(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' If |c|p < α = β, then x2 is indifferent fixed point for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3) and SI(x2) = SI(x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' If |c|p = α = β and |a − c2|p = α2, then x2 is indifferent fixed point for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3) and SI(x2) = Uα(x2), SI(x2) ∩ SI(x1) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' If |c|p = α = β and |a − c2|p < α2, then x2 is attractive fixed point for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3) and A(x2) = Uα(x2) ⊂ Sα(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' If α < β, then x2 ∈ Sβ(0) is repelling fixed point for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3) and the inequality |f(x) − x2|p > |x − x2|p holds for all x ∈ Uβ(x2), x ̸= x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let x ∈ Sr(x1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=', |x|p = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Then, from the equalities (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='2) and the properties of the p-adic norm, we have the following |f(x)|p = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 r, if r < α, ≥ α, if α ≤ r ≤ β, |a|p r , if r > β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' From this equality, f(Sr(x1)) ⊂ Sr(x1) for arbitrary r < α, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' we have SI(x1) = Uα(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' ERGODICITY AND PERIODIC ORBITS 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Note that |a|p = αβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' If |c|p < α = β, then |f ′(x2)|p = ���1 − c2 a ��� p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' From this x2 is indifferent fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let x ∈ Sr(x2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=', |x − x2|p = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Then from the equality |f(x) − x2|p = |x − x2|p · |x2(x − x2) + (x2 2 − a)|p |(x − x2) + ˆx1|p|(x − x2) + ˆx2|p (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3) we have |f(x) − x2|p = r for all r < α and |f(x) − x2|p ≥ r for r = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Thus, f(Sr(x2)) ⊂ Sr(x2) for arbitrary r < α, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' we have SI(x2) = Uα(x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' In this case, we have |x2|p = |c|p < α, so x2 ∈ Uα(0) = SI(x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Since these two Siegel disks have the same radii and share a common point, they are the same, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=', SI(x2) = SI(x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' If |c|p = α = β and |a − c2|p = α2, then |f ′(x2)|p = ��� a−c2 a ��� p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' From this x2 is indifferent fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' As above, from equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3) we get SI(x2) = Uα(x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' However, in this case x2 ∈ Sα(0), so SI(x2) ∩ SI(x1) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' If |c|p = α = β and |a − c2|p < α2, then |f ′(x2)|p = ��� a−c2 a ��� p < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' From this x2 is attractive fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Note that |x2|p = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let x ∈ Uα(x2) ⊂ Sα(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Then from equality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3) and using the strong triangle inequality of the p-adic norm we derive the relation |f(x) − x2|p < |x − x2|p for all x ∈ Uα(x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Similarly, if x /∈ Uα(x2), then we have the relation |f(x) − x2|p ≥ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Note that, the set of valuations of p-adic norm is {pm| m ∈ Z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Thus, the relation |f(x) − x2|p < |x − x2|p is equivalent to the relation |f(x) − x2|p ≤ 1 p|x − x2|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' This means that the map f : Uα(x2) → Uα(x2) is a contraction map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' According to the properties of contraction map, we have the equality A(x2) = Uα(x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' If α < β, then we have |x2|p = β, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=', x2 ∈ Sβ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Also, |f ′(x2)|p = ���1 − c2 a ��� p = β α > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let x ∈ Sr(x2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=', |x − x2|p = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Then from the equality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3) we get |f(x) − x2|p = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 β α|x − x2|p, if r < α, ≥ β, if r = α, β, if α < r < β, ≤ β, if r = β, β, if r > β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' From this we conclude that the inequality |f(x)−x2|p > |x−x2|p is holds for all x ∈ Uβ(x2), x ̸= x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' □ Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' • The spheres Sr(x1) is invariant for f if and only if r < α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' The spheres Sr(x2) is invariant for f if and only if one of the statements holds a) |c|p < α = β and r < α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' b) |c|p = α = β, |a − c2|p = α2 and r < α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' 6 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' SATTAROV, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' ALIEV 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Ergodicity of the dynamical systems on invariant spheres Recall that an invariant measure is a measure that is preserved by some function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' In ergodic theory of dynamical systems an invariant measure is very important .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let G be a topological group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' If G is abelian and locally compact, then it is well known [7] that it has a nonzero translation-invariant measure µ, which is unique up to scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' This is called the Haar measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' In the field of p-adic numbers let Σ be the minimal σ-algebra containing all open and closed (clopen) subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' A measure µ(Vρ) = ρ, Vρ ∈ Σ is usually called a Haar measure, where Vρ is a ball with radius ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' However, in some cases, the problem of studying the dynamical system of a function that mapping a compact subset of Qp to itself arises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' At this time, is needed a measure defined on σ-algebra with the unit a compact set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' If this compact set has some algebraic structure, then can we look at the natural Haar measure?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' If the considered compact set is a ball or a sphere, the answer to this question is positive, which is given as follows in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let Vr(a) be the ball (Sr(a) be the sphere) with the center at the point a ∈ Qp and B is the algebra generated by clopen subsets of Vr(a) (Sr(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' It is known that every element of B is a union of some balls Vρ(s) ⊂ Vr(a), s ∈ Vr(a) (Vρ(s) ⊂ Sr(a), s ∈ Sr(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' [16] A measure ¯µ : B → pZ is a Haar measure if it is defined by ¯µ(Vρ(s)) = ρ for all Vρ(s) ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Also, ergodic theory often deals with ergodic transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Here is the definition: Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' [18] Let T : X → X be a measure-preserving transformation on a measure space (X, Σ, µ), with µ(X) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Then T is ergodic if for every E in Σ with T −1(E) = E, either µ(E) = 0 or µ(E) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' In this section we are interested in ergodicity (with respect to Haar measure) of the dynamical systems on invariant spheres with the center at the fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='. Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Corollary 1 in the previous section gives a classification of invariant spheres centered at a fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Also, in part 2 of Theorem 1, it is proved that maximal Siegel discs consisting of union of invariant spheres fall on top of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Therefore, the center of invariant spheres is not significant when |c|p < α = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' However, when |c|p = α = β, it is necessary to consider separately the ergodicity of dynamical systems in invariant spheres with centers x1 and x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' For each invariant sphere we consider a measurable space (Sr(xi), B), here B is the algebra generated by closed subsets of Sr(xi), i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Every element of B is a union of some balls Vρ(s) ⊂ Sr(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' A measure ¯µ : B → R is a Haar measure if it is defined by ¯µ(Vρ(s)) = ρ for all s ∈ Sr(xi) and ρ ∈ pZ such that Vρ(s) ⊂ Sr(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Note that Sr(xi) = Vr(xi) \\ V r p (xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' So, we have ¯µ(Sr(xi)) = r(1 − 1 p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' ERGODICITY AND PERIODIC ORBITS 7 We consider normalized (probability) Haar measure: µ(Vρ(s)) = ¯µ(Vρ(s)) ¯µ(Sr(xi)) = pρ (p − 1)r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let Sr(xi), i = 1, 2 be invariant sphere for the function f given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Then the function f : Sr(xi) → Sr(xi) is an isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' By the Corollary 1, if the sphere Sr(xi), i = 1, 2 is invariant for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3), then r < α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' From relation x, y ∈ Sr(x1) we have |x|p = |y|p = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Then, we get the following |f(x) − f(y)|p = |x − y|p · |a|p|a − xy|p |(x − ˆx1)(x − ˆx2)(y − ˆx1)(y − ˆx2)|p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='1) Note that |a|p = αβ and |x|p = |y|p = r < α ≤ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Then, |f(x) − f(y)|p = |x − y|p · α2β2 α2β2 = |x − y|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Consequently, the function f : Sr(x1) → Sr(x1) is an isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let i = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Then by Corollary 1 we have two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' If |c|p < α = β , then by Remark 2, this case overlaps with case i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' If |c|p = α = β and |a − c2|p = α2, then by part 3 of Theorem 1, we have the relation Sr(x2) ⊂ Sα(0) for all invariant sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' So, we have |x − x2|p = r < α and |x|p = α for all x ∈ Sr(x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let x, y ∈ Sr(x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Then |f(x) − f(y)|p = |x − y|p · |a|p|(a − x2 2) + x2(x2 − y) + y(x2 − x)|p |[(x − x2) + ˆx1][(x − x2) + ˆx2][(y − x2) + ˆx1][(y − x2) + ˆx2]|p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Note that |a|p = α2, |x − x2|p = |y − x2|p = r < α and |a − x2 2|p = |a − c2|p = α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Then, |f(x) − f(y)|p = |x − y|p · α4 α4 = |x − y|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Consequently, the function f : Sr(x2) → Sr(x2) is an isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' □ Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let the conditions of the above theorem be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Then f : Sr(xi) → Sr(xi), i = 1, 2 is a measure-preserving transformation on a measure space (Sr(xi), B, µ), where µ is a normalized Haar measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' In [16], given an important results about the dynamics of isometric maps, and since the function (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3) we are considering is also an isometry, the results obtained in [16] are also relevant for the dynamics of the function (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=', if Sr(xi), i = 1, 2 is invariant sphere for the function f given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3), then we have the following: The function f : Sr(xi) → Sr(xi), i = 1, 2 is bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' For any initial point x ∈ Sr(xi), i = 1, 2 (except fixed point) the orbit {f n(x)| n ∈ N} isn’t convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' The result of the following Lemma is given as a condition in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let Sr(xi), i = 1, 2 be invariant sphere for the function f given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3), then we denote ρ(r, x) = |f(x) − x|p for x ∈ Sr(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' 8 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' SATTAROV, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' ALIEV Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' If r ̸= |c|p, then for the function f given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3) the value ρ(r, x) does not depend to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' We consider all cases in Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Then r < α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' By simple calculation we get ρ(r, x) = ���� ax x2 + cx + a − x ���� p = |x|2 p · |x + c|p |x − ˆx1|p|x − ˆx2|p = \uf8f1 \uf8f2 \uf8f3 r2|c|p αβ , if r < |c|p, r3 αβ, if r > |c|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let i = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' In this case, according to Remark 2, it is sufficient to prove the Lemma when |c|p = α = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' So, we have r = |x − x2|p = |x + c|p < α and ρ(r, x) = ���� ax x2 + cx + a − x ���� p = |x + c|p · |(x + c) − c|2 p |(x + c) + ˆx1|p|(x + c) + ˆx2|p = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' □ So, we denote ρ(r) = |f(x) − x|p for all x ∈ Sr(xi), i = 1, 2, r ̸= |c|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' In that case, we have the following assertions from [16]: The ball with radius ρ(r) is minimal invariant ball for f : Sr(xi) → Sr(xi), i = 1, 2, r ̸= |c|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let µ be normalized Haar measure on Sr(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Then a) the dynamical system (Sr(xi), f, µ) is not ergodic for all p ≥ 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' b) the dynamical system (Sr(xi), f, µ) may be ergodic if and only if r = 2ρ(r) for p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Then according to the above the dynamical system (Sr(x2), f, µ) is not ergodic, because r = ρ(r) for i = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' If i = 1, then x1 = 0 and we consider the dynamical system (Sr(0), f, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Recall Z2 = {x ∈ Q2 : |x|2 ≤ 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' So we have 1 + 2Z2 = S1(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' The following theorem gives a criterion of ergodicity for the rational functions mapping S1(0) to itself: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' [9] Let f, g : 1 + 2Z2 → 1 + 2Z2 be polynomials whose coefficients are 2-adic integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Set f(x) = � i aixi, g(x) = � i bixi, and A1 = � i odd ai, A2 = � i even ai, B1 = � i odd bi, B2 = � i even bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' The rational function R = f g is ergodic if and only if one of the following situations occurs: (1) A1 = 1(mod 4), A2 = 2(mod 4), B1 = 0(mod 4) and B2 = 1(mod 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' (2) A1 = 3(mod4), A2 = 2(mod 4), B1 = 0(mod 4) and B2 = 3(mod 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' (3) A1 = 1(mod 4), A2 = 0(mod 4), B1 = 2(mod 4) and B2 = 1(mod 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' (4) A1 = 3(mod 4), A2 = 0(mod 4), B1 = 2(mod 4) and B2 = 3(mod 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' (5) One of the previous cases with f and g interchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' ERGODICITY AND PERIODIC ORBITS 9 Consider x = g(t) = r−1t for t ∈ S1(0), then g−1 ◦ f ◦ g : S1(0) → S1(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let B (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' B1) be the algebra generated by closed subsets of Sr(0) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' S1(0)), and µ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' µ1) be normalized Haar measure on B (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' [14] The dynamical system (Sr(0), f, µ) is ergodic if and only if (S1(0), g−1 ◦ f ◦ g, µ1) is ergodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Now using the above mentioned results for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3) when p = 2 and we prove the following Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Then the dynamical system (Sr(0), f, µ) is ergodic if and only if |c|2 = β and r = α 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let r = 2l, α = 2m, β = 2k and |c|2 = 2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Since α ≤ β we have m ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Also, since c = −ˆx1 − ˆx2 and a = ˆx1ˆx2 we have q ≤ k and |a|2 = 2m+k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Note that the sphere S2l(0) is invariant for f iff l < m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' We consider the function g : S1(0) → Sr(0) defined by x = g(t) = 2−lt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Note that the function g−1(f(g(t))) : S1(0) → S1(0) has the following form g−1(f(g(t))) = t 2−2l a t2 + 2−lc a t + 1 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='2) for the function f given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Note that k, l, m, q ∈ Z, l < m ≤ k and q ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' So we have the inequalities l − m ≤ −1 and l − k ≤ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' In (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='2) we can easily see the following ���� 2−2l a t2 ���� 2 = 22l−(m+k) ≤ 2−2, ���� 2−lc a t ���� 2 = 2l+q−(m+k) ≤ 2−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Consequently, t =: γ1(t), is such that γ1 : 1 + 2Z2 → 1 + 2Z2 and 2−2l a t2 + 2−lc a t + 1 =: γ2(t) is such that γ2 : 1 + 2Z2 → 1 + 2Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Hence the function (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='2) satisfies all condition of Theorem 4, therefore using this theorem, we get A1 = 1, A2 = 0, B1 = 2−lc a and B2 = 2−2l a + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Moreover, A1 = 1(mod 4), A2 = 0(mod 4), B1 ∈ 2m+k−(l+q)(1 + 2Z2) and B2 = 1(mod 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' By these relations and Theorem 4 we get m+k−(l+q) = (m−l)+(k−q) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Note that l < m and q ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Therefore we conclude that the dynamical system (S1(0), g−1 ◦ f ◦ g, µ1) is ergodic if and only if q = k and l = m − 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=', |c|2 = β and r = α 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Consequently, by Theorem 5, (Sr(0), f, µ) is ergodic if and only if |c|2 = β and r = α 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' □ 10 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' SATTAROV, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' ALIEV 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Periodic orbits In this section we are interested in periodic trajectories and their characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Since our function is an isometry on an invariant sphere, we get the following result about periodic trajectories from [16]: Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' If the dynamical system (Sr(xi), f), i = 1, 2 has n-periodic orbit y0 → y1 → .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' → yn → y0, then the following statements hold: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' yk ∈ Vρ(r)(y0) for all k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=', n};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Character of periodic points is indifferent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' If ρ ≤ ρ(r), then we have f(Sρ(yk)) ⊂ Sρ(yk+1) for any k ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='n − 1} and f(Sρ(yn)) ⊂ Sρ(y0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Now we prove the following theorems about the existence of 2-periodic and 3-periodic trajectories: Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' If √ c2 − 2a ∈ Qp, then the function (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3) has unique 2-periodic orbit {t1, t2}, where t1,2 = −c ± √ c2 − 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' We consider the equation f 2(x) − x f(x) − x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Then we obtain the following (x2 + 2cx + 2a)(x2 + cx + a) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Since x2 + cx + a ̸= 0, we get x2 + 2cx + 2a = 0, and t1,2 = −c ± √ c2 − 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' □ Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let Sr(xi), i = 1, 2 be invariant sphere for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3) and assume that the param- eter a ∈ Sr(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Then the function (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3) has 3-periodic orbit � a, f(a), f 2(a) � if and only if (a, c) ∈ � (h(q), qh(q) − 1) : q ∈ Qp \\ � 0, −1, −2 3 � , |h(q)|p = r � , for i = 1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='1) (a, c) ∈ � (h(q), qh(q) − 1) : q ∈ Qp \\ � 0, −1, −2 3 � , |h(q)(q + 1) − 1|p = r � , for i = 2, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='2) where h(q) = 3q2+2q 6q3+11q2+6q+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' We consider the equation f 3(x) − x f(x) − x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' By simplifying this equation, we get the following equation P(x) = x6 + 6cx5 + (11c2 + 6a)x4 + (6c3 + 20ac)x3 + (15ac2 + 9a2)x2 + 12a2cx + 3a3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' ERGODICITY AND PERIODIC ORBITS 11 Necessity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let a ∈ Sr(xi) be a 3-periodic point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Then P(a) = 0 and from this we have the equality a3 + 6(c + 1)a2 + (11c + 9)(c + 1)a + 3(2c + 1)(c + 1)2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3) According to equality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3), since a ̸= 0, we have c ̸= −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Denote q = c + 1 a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Then by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3) we get (6q3 + 11q2 + 6q + 1)a − (3q2 + 2q) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' If we denote a := h(q) = 3q2 + 2q 6q3 + 11q2 + 6q + 1, then c = qh(q) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Notice that h(q) is undefined at q = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Applying the conditions that a(c + 1) ̸= 0 we see that q ̸= 0 and q ̸= − 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' For i = 1, we have |a|p = |h(q)|p = r, analogically for i = 2 we have |a + c|p = |h(q)(q + 1) − 1|p = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Summarizing the above, we get (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='1) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Sufficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Let conditions (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='1) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='2) be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Then it is easy to see that P(a) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Hence, a ∈ Sr(xi) is 3-periodic point for f given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Availability of data The datasets supporting the conclusions of this article are included in the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Acknowledgements We thank our supervisor U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE1T4oBgHgl3EQfEgNA/content/2301.02890v1.pdf'} +page_content=' Rozikov for the useful discussions.' metadata={'source': 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