Rational map ax+1/x on the projective line over $\mathbb{Q}_2$

TL;DR

This paper fully characterizes the dynamical behavior of the rational map ax+1/x on the projective line over the 2-adic numbers, revealing detailed structure specific to the field Q2.

Contribution

It provides a complete description of the dynamical structure of the map on the projective line over Q2, a novel analysis in 2-adic dynamics.

Findings

Complete dynamical classification over Q2

Identification of fixed points and cycles

Description of invariant sets and measures

Abstract

The dynamical structure of the rational map $ax+1/x$ on the projective line over the field Q2 of $2$-adic numbers, is fully described.

Equations

$$\varphi (x)=ax+\frac{1}{x},a\in {\mathbb{Q}}_p.$$

$$D(x_0,r):=\{x\in {\mathbb{Q}}_p:|x-x_0{|}_p\le r\}$$

$$S(x_0,r):=\{x\in {\mathbb{Q}}_p:|x-x_0{|}_p=r\}$$

$$\varphi (x)=ax+\frac{1}{x}=\frac{a{x}^2+1}{x}.$$

$${\mathbb{P}}^1({\mathbb{Q}}_2)=\mathcal{P}⨆\mathcal{M}⨆\mathcal{B}$$

$$\varphi (x)=ax+\frac{1}{x},a\in {\mathbb{Q}}_2,$$

$$\forall x\in {\mathbb{Q}}_2,\underset{n\to \infty }{\lim }{\varphi }^n(x)=\infty .$$

$${\mathcal{J}}_{\varphi }=\left\{x\in {\mathbb{Q}}_2:|{\varphi }^n(x){|}_2=\frac{1}{\sqrt{|a{|}_2}}for\; alln\ge 0\right\}$$

$$\forall x\in {\mathcal{F}}_{\varphi },\underset{n\to \infty }{\lim }{\varphi }^n(x)=\infty .$$

$${\mathcal{F}}_{\varphi }=\bigcup _{i=0}^{\infty }{f}^{-i}(X_a)$$

$${\mathcal{J}}_{\varphi }=\left\{x\in {\mathbb{P}}^1({\mathbb{Q}}_2):{\varphi }^n(x)\notin X_{a}for\; alln\ge 0\right\}$$

$$A=\{\begin{array}{l}\left(\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\\ 1& 1& 0& 1\end{array}\right),if\; |a{|}_2<1/4,\\ \left(\begin{array}{ccccc}0& 0& 1& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 1& 1\\ 1& 1& 0& 0& 0\end{array}\right),if\; |a{|}_2=1/4.\end{array}$$

$${\mathcal{J}}_{\varphi }=\{0,\infty \}\bigcup _{k\ge 0}(S(0,2^{2k+1})\cup S(0,2^{-2k-1})).$$

$$\{\begin{array}{l}{\varphi }^{k-1}(x)\in S(0,2^3),ifx\in S(0,2^{2k+1}),\\ {\varphi }^k(x)\in S(0,2^3),ifx\in S(0,2^{-(2k+1)});\end{array}$$

$$A=\left(\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\\ 1& 1& 0& 0\end{array}\right).$$

$$x=\sum _{n=v_p(x)}^{\infty }{a}_n{p}^n(v_p(x)\in \mathbb{Z},a_n\in \{0,1,2,\cdots ,p-1\}and{a}_{v_p(x)}\ne 0).$$

$$[x_1:x_2]=[\lambda x_1:\lambda x_2].$$

$$\left\{[x:1]\in {\mathbb{P}}^1({\mathbb{Q}}_p)\mid x\in {\mathbb{Q}}_p\right\}.$$

$$\rho (P,Q)=\frac{|x_1{y}_2-x_2{y}_1{|}_p}{\max \{|x_1{|}_p,|y_1{|}_p\}\max \{|x_2{|}_p,|y_2{|}_p\}}.$$

$$\rho (z_1,z_2)$$

$$\rho (z,\infty )$$

$$a=2^{v_2(a)}(a_0+a_1\cdot 2+a_2\cdot 2^2+\cdots )$$

$${\mathbb{Q}}_2(\sqrt{-1}),{\mathbb{Q}}_2(\sqrt{2}),{\mathbb{Q}}_2(\sqrt{-2}),{\mathbb{Q}}_2(\sqrt{3}),{\mathbb{Q}}_2(\sqrt{-3}),{\mathbb{Q}}_2(\sqrt{-6})and{\mathbb{Q}}_2(\sqrt{6}).$$

$$\forall x=x_0{x}_1{x}_2\cdots \in {\Sigma }_m,(\sigma (x){)}_i=x_{i+1}.$$

$${\Sigma }_A=\left\{(x_j{)}_{j\ge 0}\in {\Sigma }_m:A_{x_i{x}_{i+1}}=1,\forall i\ge 0\right\}.$$

$${\mathcal{K}}_{f,X}:=\bigcap _{n=0}^{\infty }{f}^{-n}(X).$$

$$\forall x,y\in D(c_i,p^{-\tau }),|f(x)-f(y){|}_p=p^{{\tau }_i}|x-y{|}_p.$$

$$I_i:=\{j\in I:D_j\cap f(D_i)\ne \varnothing \}=\{j\in I:D_j\subset f(D_i)\},$$

$$A_{ij}=1\text{mbox}\mathrm{if}j\in I_i;A_{ij}=0\text{mbox}\mathrm{otherwise}.$$

$$\forall x,y\in D(c_i,p^{-{\tau }_i}),|f(x)-f(y){|}_p=p^{{\gamma }_i}|x-y{|}_p.$$

$$g(x)=\left\{\begin{array}[]{ll}c_{i},&\mbox{if $x=c_{i}$ for some $i\in I$;}\\ c_{i}+p^{\tau-\tau_{i}}(x-c_{i}),&\mbox{if $x\in D(c_{i},{p^{-\tau_{i}}})$ for some $i\in I$.}\end{array}\right.$$

Full text

The research of S. L. FAN on this project has received funding from the European Research Council(ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement No 647133(ICHAOS). It has also been supported by NSF of China (Grant No.s 11401236 and 11471132)

1. Introduction

For a prime number $p$, let $\mathbb{Q}_{p}$ be the field of $p$-adic numbers and $\mathbb{P}^{1}(\mathbb{Q}_{p})$ be the projective line over $\mathbb{Q}_{p}$. Recently, dynamical systems on $\mathbb{Q}_{p}$ and $\mathbb{P}^{1}(\mathbb{Q}_{p})$ have been attracting much attention. See, for example, [1, 3, 20] and their bibliographies therein.

As first natural examples, polynomials and rational maps of coefficients in $\mathbb{Q}_{p}$ have been widely studied. The dynamical structure of polynomials of coefficients in the ring $\mathbb{Z}_{p}$ of $p$-adic integers is usually described by showing all the minimal subsystems. One can find such dynamical structure results for square mapping on $\mathbb{Z}_{p}$ in [12], for quadratic polynomials on $\mathbb{Z}_{2}$ in [7] and for Chebyshev polynomials on $\mathbb{Z}_{2}$ in [13]. With respect to the spherical metric on $\mathbb{Q}_{p}$, rational maps of good reduction with degree at least $2$ have similar dynamical structure [10]. On the other hand, the polynomial ${x^{p}-x\over p}$ on $\mathbb{Z}_{p}$ is proved to be topologically conjugate to the full shift on the symbolic space with $p$ symbols and thus exhibits chaos ([23]). This chaotic property has also been studied for quadratic polynomials in $\mathbb{Q}_{p}$ ([22, 6]) and for some general expanding polynomials in $\mathbb{Q}_{p}$ ([8]).

In [9], the dynamical structure of rational maps of degree one on $\mathbb{P}^{1}(\mathbb{Q}_{p})$ is completely depicted. However, the dynamical behaviors of higher degree rational maps are far from clear. In [14], we have started an attempt with a family of rational maps of degree $2$. That is

[TABLE]

Except for one subcase ($|a|_{p}<1,\sqrt{a}\in\mathbb{Q}_{p}$), the global picture of the dynamical structure of $\phi$ on $\mathbb{P}^{1}(\mathbb{Q}_{p})$ for $p\geq 3$ was given in [14]. In this paper, we deal with the case $p=2$ and we successfully show the global dynamical structure of $\phi$ for all cases. We remark that the case $p=2$ is usually different to the case $p\geq 3$ and has much more applications in computer science.

To describe the dynamical structure of a dynamical system, one may decompose the space into Fatou set and Julia set, on which the dynamical behaviors are quite different. Let $\phi:X\to X$ be a continuous map from a metric space to itself. The Fatou set $\mathcal{F}_{\phi}$ of $\phi$ is the set of point $z\in X$ where the iteration family $\{\phi^{n}\}_{n\geq 1}$ is equicontinuous on some neighborhood of $z$. The Julia set $\mathcal{J}_{\phi}$ is the complement of the Fatou set in $X$.

We denote by $|\cdot|_{p}$ the $p$-adic absolute value on $\mathbb{Q}_{p}$. For $x_{0}\in\mathbb{Q}_{p}$ and $r>0$, let

[TABLE]

be the closed disk centered at $x_{0}$ with radius $r$, and let

[TABLE]

be the sphere centered at $x_{0}$ with radius $r$. The projective line $\mathbb{P}^{1}(\mathbb{Q}_{p})$ which can be viewed as $\mathbb{Q}_{p}\cup\{\infty\}$ is equipped the spherical metric (see Section 2).

With respect to the spherical metric, a rational map $\phi\in\mathbb{Q}_{p}(z)$ induces a continuous dynamical system from $\mathbb{P}^{1}(\mathbb{Q}_{p})$ to itself, denoted by $(\mathbb{P}^{1}(\mathbb{Q}_{p}),\phi)$. If the rational map $\phi$ has good reduction (see the definition in [20], page 58, see also [10]), then it is $1$-Lipschitz continuous on $\mathbb{P}^{1}(\mathbb{Q}_{p})$ and thus has empty Julia set. A structure theorem for good reduction maps with degree at least $2$ has already been obtained in [10].

Observe that

[TABLE]

In all cases, $\infty$ is a fixed point. If $p=2$ and $|a|_{2}=1$, the map $\phi$ has good reduction which implies $\mathcal{J}_{\phi}=\emptyset$. Further, by Theorem 1.2 of [10], we have the following minimal decomposition.

Theorem 1.1**.**

Let $\phi(x)=ax+\frac{1}{x},a\in\mathbb{Q}_{2}$. If $|a|_{2}=1$, the dynamical system $(\mathbb{P}^{1}(\mathbb{Q}_{2}),\phi)$ can be decomposed into

[TABLE]

where $\mathcal{P}$ is the finite set consisting of all periodic points of $\phi$, $\mathcal{M}=\bigsqcup_{i}\mathcal{M}_{i}$ is the union of all (at most countably many) clopen invariant sets such that each $\mathcal{M}_{i}$ is a finite union of balls and each subsystem $\phi:\mathcal{M}_{i}\to\mathcal{M}_{i}$ is minimal, and each point in $\mathcal{B}$ lies in the attracting basin of a periodic orbit or of a minimal subsystem.

To complete the study of the dynamical behaviors of

[TABLE]

on $\mathbb{P}^{1}(\mathbb{Q}_{2})$, we are left two cases: $|a|_{2}>1$ and $|a|_{2}<1$. Solving these two cases is the main content of the present paper. In the rest two cases, we often find subsystems topologically conjugate to the full shift $(\Sigma_{2},\sigma)$ or some subshift of finite type $(\Sigma_{A},\sigma)$ (see the notation and definitions in Section 2).

When $|a|_{2}>1$, we have the following theorem.

Theorem 1.2**.**

If $|a|_{2}>1$, the dynamical system $(\mathbb{P}^{1}(\mathbb{Q}_{2}),\phi)$ is described as following.

• (i)

If $|a|_{2}>1$ and $\sqrt{1-a}\notin\mathbb{Q}_{2}$, then $\mathcal{F}_{\phi}=\mathbb{P}^{1}(\mathbb{Q}_{2})$. Moreover,

[TABLE]

• (ii)

If $|a|_{2}>1$ and $\sqrt{1-a}\in\mathbb{Q}_{2}$, then

[TABLE]

and $(\mathcal{J}_{\phi},\phi)$ is topologically conjugate to the full shift $(\Sigma_{2},\sigma)$. Moreover,

[TABLE]

When $|a|_{2}<1$, we set $X_{a}:=\left\{x\in\mathbb{Q}_{2}:\sqrt{|a|_{2}}<|x|_{2}<\frac{1}{\sqrt{|a|_{2}}}\right\}$. We have the following theorem.

Theorem 1.3**.**

Assume $|a|_{2}<1$. Then

[TABLE]

and we have a minimal decomposition of the subsystem $(X_{a},\phi)$ as stated in Theorem 1.1.

For the Julia set $\mathcal{J}_{\phi}$, we distinguish the following three cases.

• (i)

If $\sqrt{-a}\in\mathbb{Q}_{2}$, then

[TABLE]

and the subsystem $\left(\mathcal{J}_{\phi}\cap\left(\mathbb{P}^{1}(\mathbb{Q}_{2})\setminus(S(0,\sqrt{|a|_{2}})\cap X_{a})\right),\phi\right)$ is topologically conjugate to a subshift of finite type $(\Sigma_{A},\sigma),$ where the incidence matrix is

[TABLE]

• (ii)

If $\sqrt{-a}\notin\mathbb{Q}_{2}$, $\sqrt{-a}\in\mathbb{Q}_{2}(\sqrt{-3})$ and $|a|_{2}=1/4$, then

[TABLE]

Furthermore,

[TABLE]

and the subsystem $(S(0,2^{3})\cup S(0,2^{-3}))\cup S(0,2),\phi)$ is topologically conjugate to a subshift of finite type $(\Sigma_{A},\sigma),$ where the incidence matrix is

[TABLE]

• (iii)

Otherwise, $\mathcal{J}_{\phi}=\{0,\infty\}$.

2. Preliminaries

Consider the field $\mathbb{Q}$ of rational numbers and a prime $p\geq 2$. Any nonzero rational number $r\in\mathbb{Q}$ can be written as $r=p^{v}\frac{a}{b}$ where $v,a,b\in\mathbb{Z}$ and $(p,a)=1$ and $(p,b)=1$ (here $(x,y)$ denotes the greatest common divisor of two integers $x$ and $y$). We define $v_{p}(r)=v$ and $|r|_{p}=p^{-v_{p}(r)}$ for $r\not=0$ and $|0|_{p}=0$. Then $|\cdot|_{p}$ is a non-Archimedean absolute value on $\mathbb{Q}$. That means

(i)   $|r|_{p}\geq 0$ with equality only for $r=0$;

(ii)  $|rs|_{p}=|r|_{p}|s|_{p}$;

(iii) $|r+s|_{p}\leq\max\{|r|_{p},|s|_{p}\}$.

The field of $p$-adic numbers $\mathbb{Q}_{p}$ is the completion of $\mathbb{Q}$ under the absolute value $|\cdot|_{p}$. Actually, any $x\in\mathbb{Q}_{p}$ can be written uniquely as

[TABLE]

Here, the integer $v_{p}(x)$ is called the $p$-valuation of $x$.

Any point in the projective line $\mathbb{P}^{1}(\mathbb{Q}_{p})$ may be given in homogeneous coordinates by a pair $[x_{1}:x_{2}]$ of points in $\mathbb{Q}_{p}$ which are not both zero. Two such pairs are equal if they differ by an overall (nonzero) factor $\lambda\in\mathbb{Q}_{p}^{*}$:

[TABLE]

The field $\mathbb{Q}_{p}$ may be identified with the subset of $\mathbb{P}^{1}(\mathbb{Q}_{p})$ given by

[TABLE]

This subset contains all points in $\mathbb{P}^{1}(\mathbb{Q}_{p})$ except one: the point of infinity, which may be given as $\infty=[1:0].$

The spherical metric defined on $\mathbb{P}^{1}(\mathbb{Q}_{p})$ is analogous to the standard spherical metric on the Riemann sphere. If $P=[x_{1}:y_{1}]$ and $Q=[x_{2}:y_{2}]$ are two points in $\mathbb{P}^{1}(\mathbb{Q}_{p})$, we define

[TABLE]

Viewing $\mathbb{P}^{1}(\mathbb{Q}_{p})$ as $\mathbb{Q}_{p}\cup\{\infty\}$, the spherical distance of $z_{1},z_{2}\in\mathbb{Q}_{p}\cup\{\infty\}$, can be described by

[TABLE]

Remark that the restriction of the spherical metric on the ring $\mathbb{Z}_{p}:=\{x\in\mathbb{Q}_{p},|x|_{p}\leq 1\}$ of $p$-adic integers is same as the metric induced by the absolute value $|\cdot|_{p}$.

In what follows, we restrict ourselves to the case $p=2$ only. We recall the conditions under which a number in $\mathbb{Q}_{2}$ has a square root in $\mathbb{Q}_{2}$, then we present all possible quadratic extensions of $\mathbb{Q}_{2}$.

Lemma 2.1** ([18]).**

Let $a$ be a nonzero $2$-adic number with its $2$-adic expansion

[TABLE]

*where $a_{0}=1$ and $0\leq a_{j}\leq 1\ (j\geq 1)$. The equation $x^{2}=a$ has a solution $x\in\mathbb{Q}_{2}$ if and only if $v_{p}(a)$ is even and $a_{1}=a_{2}=0$. *

Lemma 2.2** ( [18, Theorem 1, p.72]).**

For $a,a^{\prime}\in\mathbb{Q}_{2}\setminus\{0\}$, $\mathbb{Q}_{2}(\sqrt{a})=\mathbb{Q}_{2}(\sqrt{a^{\prime}})$ if and only if the quotient $a/a^{\prime}$ is a square of a $2$-adic number.

There are exactly $7$ distinct quadratic extensions of $\mathbb{Q}_{2}$ which are represented respectively by

[TABLE]

Now we recall some standard terminology of the theory of dynamical systems.

A point $x_{0}\in X$ is called a fixed point of $\phi$ if $\phi(x_{0})=x_{0}$. For a fixed point $x_{0}\in\mathbb{Q}_{p}$, the derivative $\phi^{\prime}(x_{0})$ (if exists) is called the multiplier of $x_{0}$. Remark that the multiplier is invariant by changing of coordinate. If $\infty$ is a fixed point, then the multiplier of $\infty$ is $\psi^{\prime}(0)$, where $\psi(x)=\frac{1}{\phi(1/x)}$. A fixed point is called attracting, indifferent, or repelling accordingly as the absolute value of its multiplier is less than, equal to, or greater than $1$. Fixed points of multiplier [math] are called super attracting.

A subsystem of a dynamical system is minimal if the orbit of any point in the subspace is dense in the subspace.

For $m\geq 1$, call the finite set $\mathcal{A}:=\{0,1,\cdots,m-1\}$ an alphabet. The infinite product space $\Sigma_{m}:=\mathcal{A}^{\mathbb{N}}$ is said to be the symbolic space of $m$ symbols. The (left)-shift $\sigma$ on $\Sigma$ is defined as

[TABLE]

The couple $(\Sigma_{m},\sigma)$ is called a full shift on $m$ symbols. Let $A$ be a $m\times m$ matrix with entries in $\{0,1\}$ and set

[TABLE]

Then $\Sigma_{A}$ is an invariant subset of $\Sigma_{m}$. We thus obtain a subsystem $(\Sigma_{A},\sigma)$ called a subshift of finite type.

A large family of $p$-adic dynamical systems are conjugate to full shift or subshift of finite type. Let $f:X\to\mathbb{Q}_{p}$ be a map from a compact open set $X$ of $\mathbb{Q}_{p}$ into $\mathbb{Q}_{p}$ with $f(X)\supset X$. For such a map $f$, define

[TABLE]

It is clear that $f^{-1}(\mathcal{K}_{f,X})=\mathcal{K}_{f,X}$ and then $f(\mathcal{K}_{f,X})\subset\mathcal{K}_{f,X}$. Hence, we have a subsystem $(\mathcal{K}_{f,X},f)$. The subset $\mathcal{K}_{f,X}$ is sometimes called filled Julia set. For a complex dynamical system, the Julia set is the boundary of filled Julia set in $X$. However, it is not true for in $\mathbb{Q}_{p}$, since $\mathbb{Q}_{p}$ is not algebraically closed.

Assume that $X=\bigsqcup_{i\in I}D(c_{i},{p^{-\tau}})$ can be written as a finite disjoint union of disks of centers $c_{i}$ and of the same radius $p^{-\tau}$ (with some $\tau\in\mathbb{Z}$) such that for each $i\in I$ there is an integer $\tau_{i}\in\mathbb{Z}$ such that

[TABLE]

The subsystem $(\mathcal{K}_{f,X},f)$ is called a $p$-adic weak repeller if all $\tau_{i}$ in (2.3) are nonnegative, but at least one is positive.

The chaotic dynamical behavior of a $p$-adic weak repeller is determined by a matrix with entries in $\{0,1\}$. Denote simply the disk $D(c_{i},{p^{-\tau}})$ by $D_{i}$. For any $i\in I$, let

[TABLE]

and define the incidence matrix $A=(A_{i,j})_{I\times I}$ of $(\mathcal{J}_{f},f)$, by

[TABLE]

The weak repeller $(\mathcal{K}_{f,X},f)$ is called transitive if the matrix $A$ is irreducible. In [8], the authors proved the following theorem.

Theorem 2.3** ([8], Theorem 1.1).**

Let $(\mathcal{K}_{f,X},f)$ be a transitive $p$-adic weak repeller in $\mathbb{Q}_{p}$ with incidence matrix $A$. Then $(\mathcal{K}_{f,X},f)$ is topologically conjugate to $(\Sigma_{A},\sigma)$, i.e., there exists a continuous bijection $h:\mathcal{K}_{f,X}\rightarrow\Sigma_{A}$ such that $h\circ f=\sigma\circ h$. Moreover, the Julia set of the system $(\mathcal{K}_{f,X},f)$ is the whole space $\mathcal{K}_{f,X}$.

More generally, we can apply this idea to “generalized $p$-adic repellers”. Assume that $f:X\to\mathbb{Q}_{p}$ is a map from a compact open set $X=\bigsqcup_{i\in I}D(c_{i},{p^{-\tau_{i}}})$ into $\mathbb{Q}_{p}$ such that $f(X)\supset X$ and for each $i\in I$ there is an integer $\gamma_{i}\in\mathbb{Z}$ such that

[TABLE]

Denote simply the disk $D(c_{i},{p^{-\tau_{i}}})$ by $D_{i}$. The subsystem $(\mathcal{K}_{f,X},f)$ is called a generalized $p$-adic weak repeller if the map $f:X\to\mathbb{Q}_{p}$ satisfies the following two conditions:

(i) for each $i\in I$, $f(D_{i})$ contains at least some $D_{j}$;

(ii) there exists at least one $i\in I$ such that $f(D_{i})\supsetneq D_{j}$.

We define similarly the incidence matrix $A$ associated to $(\mathcal{K}_{f,X},f)$. The generalized $p$-adic weak repeller $(\mathcal{K}_{f,X},f)$ is transitive if $A$ is transitive.

Theorem 2.4**.**

Let $(\mathcal{K}_{f,X},f)$ be a transitive generalized $p$-adic weak repeller in $\mathbb{Q}_{p}$ with incidence matrix $A$. Then $(\mathcal{K}_{f,X},f)$ is topologically conjugate to $(\Sigma_{A},\sigma)$, i.e., there exists a continuous bijection $h:\mathcal{K}_{f,X}\rightarrow\Sigma_{A}$ such that $h\circ f=\sigma\circ h$. Moreover, the Julia set of the system $(\mathcal{K}_{f,X},f)$ is the whole space $\mathcal{K}_{f,X}$.

Proof.

Let $\tau=\max_{i\in I}\{\tau_{i}\}$ and $X^{g}=\bigsqcup_{i\in I}D(c_{i},{p^{-\tau}})$. Take a map $g:X\to X^{g}$ such that

[TABLE]

Then the system $(X,f)$ is topologically conjugate to $(X^{g},f^{g}=g\circ f\circ g^{-1})$. Note that $X^{g}$ consists of ball of same the radius $p^{-\tau}$. The condition (i) implies that $f^{g}$ is non contracting on each ball $g(D_{i})$ of $X^{g}$ and the condition (ii) implies $f^{g}$ is expanding on at least one ball of $X^{g}$. Observing that the associated incidence matrix does not change by the topological conjugacy $g$, we conclude by applying Theorem 2.3. ∎

3. Dynamical systems

Recall that we need to study the rational map $\phi(x)=ax+\frac{1}{x},a\in\mathbb{Q}_{2}$ for the rest cases $|a|_{2}>1$ and $|a|_{2}<1$. An easy computation shows that

[TABLE]

and

[TABLE]

3.1. $|a|_{2}>1$

We distinguish two subcases: $\sqrt{1-a}\notin\mathbb{Q}_{2}$ and $\sqrt{1-a}\in\mathbb{Q}_{2}.$

In the first subcase, every point goes to infinity.

Proposition 3.1**.**

Suppose $|a|_{2}>1$. If $\sqrt{1-a}\notin\mathbb{Q}_{2}$, then

[TABLE]

Proof.

By the assumption $|a|_{2}>1$, for all $x\in\mathbb{Q}_{2}$ such that $|x|_{2}\geq 1$, we have

[TABLE]

Thus the absolute values of the iterations $\phi^{n}(x)$ are strictly increasing. Hence

[TABLE]

That is to say, $\{x\in\mathbb{Q}_{2}:|x|_{2}\geq 1\}$ is included in the attracting basin of $\infty$.

Now we investigate the points in the ball $\{x\in\mathbb{Q}_{2}:|x|_{2}\leq 1/2\}$. We partition this ball into two:

[TABLE]

If $x\in A_{1}$, then

[TABLE]

Thus by (3.1), $\phi(x)$ falls in the attracting basin of $\infty$, and $\lim_{n\to\infty}\phi^{n}(x)=\infty$.

For $x\in A_{2}$, we will study separately according to the parity of $v_{2}(a)$.

When $v_{2}(a)$ is odd, $A_{2}=\emptyset$. So, we can conclude that

[TABLE]

When $v_{2}(a)$ is even, we distinguish two cases: (i) $v_{2}(a)=-2$ and (ii) $v_{2}(a)<-2$. Note that $A_{2}=S(0,1/\sqrt{|a|_{2}})=D(\sqrt{|a|_{2}},\frac{1}{4\sqrt{|a|_{2}}})\cup D(-\sqrt{|a|_{2}},\frac{1}{4\sqrt{|a|_{2}}})$. Let $x\in A_{2}$. Then we have $x=\sqrt{|a|_{2}}+y$ or $-\sqrt{|a|_{2}}+y$ for some $y\in D(0,\frac{1}{4\sqrt{|a|_{2}}})$. Without loss of generality, we assume $x=\sqrt{|a|_{2}}+y$ with $y\in D(0,\frac{1}{4\sqrt{|a|_{2}}})$. So we have

[TABLE]

(i) $v_{2}(a)=-2$. By Lemmas 2.1 and 2.2, the assumption $\sqrt{1-a}\notin\mathbb{Q}_{2}$ implies

[TABLE]

If $2^{-v_{2}(a)}a=1\hbox{ or }-3~{}(\!\!\!\!\mod 8),$ then

[TABLE]

If $2^{-v_{2}(a)}a=-1~{}(\!\!\!\!\mod 8),$ then

[TABLE]

So $\phi(x)\in A_{1}$ and we conclude by the above study of the points in $A_{1}$.

(ii) $v_{2}(a)$ is even and $v_{2}(a)\neq-2$. By Lemmas 2.1 and 2.2 ,$\sqrt{1-a}\notin\mathbb{Q}_{2}$ if and only if $\sqrt{-a}\notin\mathbb{Q}_{2}$ which implies $2^{-v_{2}(a)}a=1$ or $\pm 3~{}(\!\!\!\!\mod 8)$. So we have

[TABLE]

By (3.1), the proof is completed.

∎

Now we study the second subcase $\sqrt{1-a}\in\mathbb{Q}_{2}$. We first find the fixed points and then investigate the dynamical behaviors nearby the fixed points.

Lemma 3.2**.**

Suppose $|a|_{2}>1$. If $\sqrt{1-a}\in\mathbb{Q}_{2}$, then $\phi$ has two repelling fixed points

[TABLE]

Proof.

It is easy to check that $x_{1}=\frac{1}{\sqrt{1-a}}$ and $x_{2}=-\frac{1}{\sqrt{1-a}}$ are the two fixed points of $\phi$. Note that

[TABLE]

Since $|a|_{2}>1$ and $\sqrt{1-a}\in\mathbb{Q}_{2}$, by Lemma 2.1, $v_{2}(a)$ is even, which implies $|a|_{2}\geq 4$. Hence, we have

[TABLE]

Therefore, both fixed points are repelling. ∎

Lemma 3.3**.**

Suppose $|a|_{2}>1$. If $\sqrt{1-a}\in\mathbb{Q}_{2}$, then

[TABLE]

and

[TABLE]

Proof.

In Lemma 3.2, we have shown $|a|_{2}\geq 4$. Since $|x_{1}-x_{2}|_{2}=\frac{1}{2\sqrt{|a|_{2}}}$, we have $x_{2}\in D(x_{1},\frac{\sqrt{|a|_{2}}}{8}).$ Thus to prove Lemma 3.3, it suffices to show that

[TABLE]

if $x,y\in D(x_{1},\frac{1}{4\sqrt{|a|_{2}}})$ or $D(x_{2},\frac{1}{4\sqrt{|a|_{2}}})$.

Without loss of generality, we assume $x,y\in D(x_{1},\frac{1}{4\sqrt{|a|_{2}}})$. Observe that the disk $D(x_{1},\frac{1}{4\sqrt{|a|_{2}}})$ is the image of $D({1\over x_{1}},\frac{\sqrt{|a|_{2}}}{4})$ by the map $f(x)=1/x$, i.e.

[TABLE]

Hence, there are $x^{\prime},y^{\prime}\in D(0,\frac{\sqrt{|a|_{2}}}{4})$ such that

[TABLE]

So

[TABLE]

Noting that $|a|_{2}\geq 4$ and $|(x^{\prime}+y^{\prime})\sqrt{1-a}|_{2}\leq|a|_{2}/4$, we have

[TABLE]

Hence,

[TABLE]

∎

The following lemma shows that the point $\infty$ is an attracting fixed point.

Lemma 3.4**.**

Suppose $|a|_{2}>1$. If $\sqrt{1-a}\in\mathbb{Q}_{2}$, then

[TABLE]

for all $x\in\mathbb{Q}_{2}\setminus(D(x_{1},\frac{1}{4\sqrt{|a|_{2}}})\cup D(x_{2},\frac{1}{4\sqrt{|a|_{2}}})).$

Proof.

For $x\in\mathbb{Q}_{2}\setminus D(0,1/2)$, we have$|1/x|_{2}\leq 1$. Then

[TABLE]

Hence,

[TABLE]

which implies

[TABLE]

Thus, to conclude it suffices to show that

[TABLE]

By noting that $D(x_{1},\frac{1}{4\sqrt{|a|_{2}}})\cup D(x_{2},\frac{1}{4\sqrt{|a|_{2}}})=S(0,\frac{1}{\sqrt{|a|_{2}}})$, we distinguish two cases.

Case (1) $|x|_{2}<1/\sqrt{|a|_{2}}$. Then $|1/x|_{2}>|ax|_{2}$, and thus

[TABLE]

Case (2) $1>|x|_{2}>1/\sqrt{|a|_{2}}$. Then $|ax|_{2}>|1/x|_{2}>1$, and hence

[TABLE]

∎

Now we are ready to show the dynamical structure of the system for the second subcase and finish the proof of Theorem 1.2.

Proposition 3.5**.**

Suppose $|a|_{2}>1$. If $\sqrt{1-a}\in\mathbb{Q}_{2}$, then

[TABLE]

and $(\mathcal{J}_{\phi},\phi)$ is topologically conjugate to $(\Sigma_{2},\sigma)$. Moreover,

[TABLE]

Proof.

Take $X=D(x_{1},\frac{1}{4\sqrt{|a|_{2}}})\cup D(x_{2},\frac{1}{4\sqrt{|a|_{2}}})$. Consider the restriction map $\phi:X\to\mathbb{P}^{1}(\mathbb{Q}_{2})$. Define $\mathcal{K}_{\phi,X}$ as (2.1). By Theorem 2.3, the $p$-adic repeller ($\mathcal{K}_{\phi,X},\phi)$ is topologically conjugate to the full shift $(\Sigma_{2},\sigma)$ and the points which do not lie in $\mathcal{K}_{\phi,X}$ are sent to $\mathbb{P}^{1}(\mathbb{Q}_{2})\setminus X.$ By Lemma 3.4,

[TABLE]

Hence we have $\mathcal{J}_{\phi}=\mathcal{K}_{\phi,X}$. ∎

Proof of Theorem 1.2.

Combining Propositions 3.1 and 3.5, we complete the proof. ∎

3.2. $|a|_{2}<1$

We begin with two lemmas which will be useful.

Lemma 3.6**.**

If $|a|_{2}<1$, then

[TABLE]

Moreover,

[TABLE]

Proof.

Note that

[TABLE]

If $x,y\in S(0,2^{i})$ with $i>v_{2}(a)/2$, then $|a|_{2}>\frac{1}{|xy|_{2}}$. So

[TABLE]

Moreover, $|ax|_{2}<|1/x|_{2}$. Thus $|\phi(x)|_{2}=|ax+1/x|_{2}=|ax|_{2}$ which implies

[TABLE]

If $x,y\in S(0,2^{i})$ with $i\leq 0$, then $|a|_{2}<\frac{1}{|xy|_{2}}$. So

[TABLE]

Moreover, $|ax|_{2}<1\leq|1/x|_{2}$. Hence $|\phi(x)|_{2}=|ax+1/x|_{2}=|1/x|_{2}$ which implies

[TABLE]

∎

Lemma 3.7**.**

If $|a|_{2}<1$, then for all $-\lfloor(v_{2}(a)-1)/2\rfloor\leq i\leq\lfloor(v_{2}(a)-1)/2\rfloor$,

[TABLE]

and $\phi^{2}$ is $1$-Lipschitz continuous on $S(0,2^{i})\cup S(0,2^{-i})$.

Proof.

If $x\in S(0,2^{i})$ for some $-\lfloor(v_{2}(a)-1)/2\rfloor\leq i\leq 0$, then by the assumption $|a|_{2}<1$, we have

[TABLE]

Now let $x\in S(0,2^{i})$ for some $0\leq i\leq\lfloor(v_{2}(a)-1)/2\rfloor$. When $i\leq\lfloor(v_{2}(a)-1)/2\rfloor$, we have

[TABLE]

which implies

[TABLE]

Hence, the first assertion of the lemma holds.

Let us show that $\phi^{2}$ is $1$-Lipschitz continuous on $S(0,2^{i})\cup S(0,2^{-i})$ for $-\lfloor(v_{2}(a)-1)/2\rfloor\leq i\leq\lfloor(v_{2}(a)-1)/2\rfloor$. Let $x,y\in S(0,2^{i})\cup S(0,2^{-i})$. If $x\in S(0,2^{i})$ and $y\in S(0,2^{-i})$, then $|xy|_{2}=1$ and

[TABLE]

Hence, it suffices to show that for each $-\lfloor(v_{2}(a)-1)/2\rfloor\leq i\leq\lfloor(v_{2}(a)-1)/2\rfloor$,

[TABLE]

By observing that $\frac{1}{|xy|_{2}}=2^{-2i}>|a|_{2}$ and $\frac{1}{|\phi(x)\phi(y)|_{2}}=2^{2i}>|a|_{2}$, we have

[TABLE]

∎

Now we distinguish two subcases: $\sqrt{-a}\in\mathbb{Q}_{2}$ and $\sqrt{-a}\notin\mathbb{Q}_{2}$.

3.2.1. $\sqrt{-a}\in\mathbb{Q}_{2}$

Lemma 3.8**.**

If $\sqrt{-a}\in\mathbb{Q}_{2}$, then

[TABLE]

and

[TABLE]

Proof.

Note that $\phi(\pm 1/\sqrt{-a})=0$. It suffices to show that

[TABLE]

Without loss of generality, we assume $x,y\in D({1\over\sqrt{-a}},\frac{1}{4\sqrt{|a|_{2}}})$. By the same arguments in the proof of Lemma 3.3, there exist $x^{\prime},y^{\prime}\in D(0,\frac{\sqrt{|a|_{2}}}{4})$ such that

[TABLE]

and

[TABLE]

Since $x^{\prime},y^{\prime}\in D(0,\frac{\sqrt{|a|_{2}}}{4})$, we immediately get

[TABLE]

Thus

[TABLE]

∎

Proposition 3.9**.**

*Suppose $|a|_{2}<1$ and $\sqrt{-a}\in\mathbb{Q}_{2}$. Let $\Omega=\mathbb{P}^{1}(\mathbb{Q}_{2})\setminus(S(0,\sqrt{|a|_{2}})\cup X_{a})$ and consider the restriction map $\phi:\Omega\to\mathbb{P}^{1}(\mathbb{Q}_{2})$. Set $\mathcal{K}_{\phi,\Omega}=\bigcap_{n=0}^{\infty}\phi^{-n}(\Omega)$. We distinguish two cases.

(i) If $v_{2}(a)\neq 2$, then the subsystem $\left(\mathcal{K}_{\phi,\Omega},\phi\right)$ is topologically conjugate to a subshift of finite type $(\Sigma_{A},\sigma),$ where the incidence matrix is*

[TABLE]

(ii)* If $v_{2}(a)=2$, then the subsystem $\left(\mathcal{K}_{\phi,\Omega},\phi\right)$ is topologically conjugate to a subshift of finite type $(\Sigma_{A},\sigma),$ where the incidence matrix is*

[TABLE]

Proof.

Let $D_{1}=D({1\over\sqrt{-a}},\frac{1}{4\sqrt{|a|_{2}}}),$ $D_{2}=D(-{1\over\sqrt{-a}},\frac{1}{4\sqrt{|a|_{2}}})$, $D_{3}=D(0,\frac{\sqrt{|a|_{2}}}{8})$ and $D_{4}=\mathbb{P}^{1}(\mathbb{Q}_{p})\setminus D(0,\frac{4}{\sqrt{|a|_{2}}})$. By Lemma 3.8, the restricted maps $\phi:D_{1}\to D_{3}$ and $\phi:D_{2}\to D_{3}$ are both scaling bijective. One can also directly check that $\phi:D_{3}\to D_{4}$ and $\phi:D_{4}\to\phi(D_{4})=\mathbb{P}^{1}(\mathbb{Q}_{p})\setminus D(0,4\sqrt{|a|_{2}})$ are bijective.

(i) Since $|a|_{2}<4$, we have $D_{1}\cup D_{2}\subset\phi(D_{4})$. Let $X=\cup_{i=1}^{4}D_{i}$. Consider the map $\phi:X\to\mathbb{P}^{1}(\mathbb{Q}_{2})$. Set

[TABLE]

Let $g:x\mapsto 1/(x-1)$ and $\psi:=g\circ\phi\circ g^{-1}$. We study the map $\psi:g(X)\to\mathbb{P}^{1}(\mathbb{Q}_{p})$. Observe $g(X)\subset\mathbb{Z}_{p}$, and that $(\mathcal{K}_{\psi,g(X)},\psi)$ is a generalized weak repeller with incidence matrix

[TABLE]

Thus, by Theorem 2.4, $(\mathcal{K}_{\psi,g(X)},\psi)$ is topologically conjugate to the subshift of finite type $(\Sigma_{A},\sigma)$. Take $\mathcal{K}_{\phi,X}=g^{-1}(\mathcal{K}_{\psi,g(X)})$. Since $\psi=g\circ\phi\circ g^{-1}$, we deduce that $(\mathcal{K}_{\phi,X},\phi)$ is topologically conjugate to $(\Sigma_{A},\sigma).$

Noting that

[TABLE]

by Lemmas 3.6 and 3.7, we have $\phi(\Omega\setminus X)\subset X_{a}$, since $v_{2}(a)>2$. Thus we have $\mathcal{K}_{\phi,X}=\mathcal{K}_{\phi,\Omega}$.

(ii) Let $D_{5}=S(0,{4\over\sqrt{|a|_{2}}})$. Since $|a|_{2}=4$, we have $D_{5}\subset\phi(D_{4})$. By Lemma 3.6, the map $\phi:D_{5}\to D_{1}\cap D_{2}$ satisfies

[TABLE]

Let $X=\cup_{i=1}^{5}D_{i}$ and consider the map $\phi:X\to\mathbb{P}^{1}(\mathbb{Q}_{2})$. By the same argument as the case (i), we have $\mathcal{K}_{\phi,X}=\mathcal{K}_{\phi,\Omega}$ and $(\mathcal{K}_{\phi,X},\phi)$ is topologically conjugate to a subshift of finite type $(\Sigma_{A},\sigma),$ with incidence matrix

[TABLE]

∎

3.2.2. $\sqrt{-a}\notin\mathbb{Q}_{2}$

Lemma 3.10**.**

If $v_{2}(a)$ is odd, then for each $x\in\mathbb{Q}_{2}\setminus\{0\}$, there exists a positive integer $N$ such that

[TABLE]

for some $0\leq i\leq\frac{v_{2}(a)-1}{2}$.

Proof.

Note that $|\phi(x)|_{2}=|{1}/{x}|_{2}$, if $|x|_{2}<2^{-(v_{2}(a)-1)/2}$. Thus we have

[TABLE]

By Lemma 3.7, it suffices to show that the statement holds for the points in $\mathbb{P}^{1}(\mathbb{Q}_{2})\setminus D(0,2^{(v_{2}(a)-1)/2})$. In fact, if $x\in\mathbb{P}^{1}(\mathbb{Q}_{2})\setminus D(0,2^{(v_{2}(a)-1)/2})$, one can check that

[TABLE]

So there exists an integer $N$ such that

[TABLE]

By Lemma 3.7, we conclude. ∎

Lemma 3.11**.**

If $v_{2}(a)>0$ is even and $\sqrt{-a}\notin\mathbb{Q}_{2}(\sqrt{-3})$, then

[TABLE]

Proof.

For $x\in S(0,\frac{1}{\sqrt{|a|_{2}}})$, we have $x=2^{-\frac{v_{2}(a)}{2}}+y$ for some $y\in D(0,\frac{1}{2\sqrt{|a|_{2}}})$. Noting that $v_{2}(a)$ is even, we deduce from $\sqrt{-a}\notin\mathbb{Q}_{2}(\sqrt{-3})$ that

[TABLE]

Thus

[TABLE]

Therefore,

[TABLE]

∎

Lemma 3.12**.**

If $\sqrt{-a}\in\mathbb{Q}_{2}(\sqrt{-3})$, then

[TABLE]

and

[TABLE]

Proof.

Without loss of generality, we assume $x,y\in D(2^{-\frac{v_{2}(a)}{2}},\frac{1}{4\sqrt{|a|_{2}}})$. By the same arguments in the proof of Lemma 3.3, there exist $x^{\prime},y^{\prime}\in D(0,\frac{\sqrt{|a|_{2}}}{4})$ such that

[TABLE]

and

[TABLE]

Note that $\sqrt{-a}\in\mathbb{Q}_{2}(\sqrt{-3})$ implies $2^{v_{2}(a)}a\equiv 3~{}{\rm(\!\!\!\!\!\mod 8)}$. Since $x^{\prime},y^{\prime}\in D(0,\frac{\sqrt{|a|_{2}}}{4})$, we immediately get

[TABLE]

Hence

[TABLE]

∎

Lemma 3.13**.**

Assume that $v_{2}(a)$ is even. If $v_{2}(a)\neq 2$ or $\mathbb{Q}_{2}(\sqrt{-a})\neq\mathbb{Q}_{2}(\sqrt{-3})$, then for each $x\in\mathbb{Q}_{2}\setminus\{0\}$, there exists a positive integer $N$ such that

[TABLE]

for some $0\leq i\leq\frac{v_{2}(a)}{2}-1$.

Proof.

If $|x|_{2}\leq\sqrt{|a|_{2}}$, then

[TABLE]

Thus we have

[TABLE]

By Lemma 3.7, it suffices to show that the statement holds for all points in $\mathbb{P}^{1}(\mathbb{Q}_{2})\setminus D(0,\frac{1}{2\sqrt{|a|_{2}}})$. In fact, if $x\in\mathbb{P}^{1}(\mathbb{Q}_{2})\setminus D(0,\frac{1}{\sqrt{|a|_{2}}})$, by noting $|a|_{2}\leq 1/16$, one can check that

[TABLE]

Hence there exists an integer $N$ such that

[TABLE]

Therefore, it remains to show that the statement holds for all $x\in S(0,\frac{1}{\sqrt{|a|_{2}}})$. We distinguish two cases.

Case (1) $v_{2}(a)\neq 2$. By Lemma 3.12, we have

[TABLE]

Observing $\phi(S(0,\frac{\sqrt{|a|_{2}}}{4}))\subset S(0,\frac{4}{\sqrt{|a|_{2}}})$ and $\phi(S(0,\frac{4}{\sqrt{|a|_{2}}}))\subset S(0,4\sqrt{|a|_{2}})$, we obtain

[TABLE]

Since $v_{2}(a)\neq 2$ , we have

[TABLE]

Hence, we conclude by Lemma 3.7 again.

Case (2) $\sqrt{-a}\in\mathbb{Q}_{2}(\sqrt{-1})$ or $\mathbb{Q}_{2}(\sqrt{3})$. By Lemma 3.11, we have

[TABLE]

Since $\phi(S(0,\frac{\sqrt{|a|_{2}}}{2}))\subset S(0,\frac{2}{\sqrt{|a|_{2}}})$ and $\phi(S(0,\frac{2}{\sqrt{|a|_{2}}}))\subset S(0,2^{-\frac{v_{2}(a)}{2}+1})$, we have

[TABLE]

Applying Lemma 3.7, we finish the proof. ∎

Proposition 3.14**.**

Assume that $|a|_{2}<1$ and $\sqrt{-a}\notin\mathbb{Q}_{2}$. If $\sqrt{-a}\in\mathbb{Q}_{2}(\sqrt{-3})$ and $v_{2}(a)=2$, then the subsystem $(S(0,2^{3})\cup S(0,2^{-3})\cup S(0,2),\phi)$ is topologically conjugate to a subshift of finite type $(\Sigma_{A},\sigma),$ with incidence matrix

[TABLE]

Proof.

Let $D_{1}=D({1\over 2},{1\over 2})$, $D_{2}=D({-1\over 2},{1\over 2})$, $D_{3}=D(8,\frac{1}{16})$ and $D_{4}=D(\frac{1}{8},4)$. By Lemma 3.12, the restricted maps $\phi:D_{1}\to D_{3}$ and $\phi:D_{2}\to D_{3}$ are both bijective. Since $\phi(x)-\phi(y)=(a-\frac{1}{xy})(x-y)$, we have

[TABLE]

and

[TABLE]

So $\phi:D_{3}\to D_{4}$ and $\phi:D_{4}\to\phi(D_{4})=D_{1}\cup D_{2}$ are bijective.

Set $\Omega=\bigcup_{i=1}^{4}D_{i}$ and consider the restricted map $\phi:\Omega\to\mathbb{P}^{1}(\mathbb{Q}_{p})$. Take

[TABLE]

By Theorem 2.4, $(\Omega,\phi)$ is topologically conjugate to a subshift of finite type $(\Sigma_{A},\sigma)$ with incidence matrix

[TABLE]

∎

Proof of Theorem 1.3.

By Lemma 3.7, we immediately have $\phi(X_{a})\subset X_{a}$ and $X_{a}\subset\mathcal{F}_{\phi}$. By the argument of the proof of Proposition 3 in [14], we obtain a minimal decomposition for the subsystem $(X_{a},\phi)$ as stated in Theorem 1.1. It remains to show that for each $x\in\mathcal{F}_{\phi}$, there exists some positive integer $N$ such that $\phi^{N}(x)\in X_{a}$. We distinguish the following three cases.

(i) Let $\Omega=\mathbb{P}^{1}(\mathbb{Q}_{2})\setminus(S(0,\sqrt{|a|_{2}})\cup X_{a})$. By Lemmas 3.6 and 3.8, we have

[TABLE]

By Proposition 3.9,

[TABLE]

By (3.2) and the definition of $\mathcal{K}_{\phi,\Omega}$, each point $x\in\Omega\setminus\mathcal{K}_{\phi,\Omega}$ goes to $X_{a}$ by iteration of $\phi$. Thus,

[TABLE]

By Lemma 3.6, $\phi(S(0,\sqrt{|a|_{2}}))=S(0,\frac{1}{\sqrt{|a|_{2}}})\subset\Omega$. Hence,

[TABLE]

The dynamical behaviors of $(\mathcal{K}_{\phi,\Omega},\phi)$ have already been shown in Proposition 3.9.

(ii) By Proposition 3.14,

[TABLE]

By Lemma 3.6, for each integer $k\geq 1$, we have

[TABLE]

Thus,

[TABLE]

However, if

[TABLE]

by Lemma 3.6, there exists some $N$ such that $\phi^{N}(x)\in X_{a}=S(0,1)$. Noting that $\phi(0)=\infty$ is a repelling fixed point, we have

[TABLE]

The dynamical behaviors of $(S(0,2^{3})\cup S(0,2^{-3})\cup S(0,2),\phi)$ have already been shown in Proposition 3.14.

(iii) By Lemmas 3.10 and 3.13, for each $x\in\mathbb{Q}_{2}\setminus\{0\}$, there exists an integer $N$ such that

[TABLE]

for some $0\leq i\leq\lfloor(v_{2}(a)-1)/2\rfloor$. By Lemma 3.7, we have $\mathbb{Q}_{p}\setminus\{0\}\in\mathcal{F}_{\phi}$. Noting that $\phi(0)=\infty$ is a repelling fixed point, we have $\mathcal{J}_{\phi}=\{0,\infty\}.$ ∎