Hyers-Ulam stability of elliptic M\"obius difference equation

TL;DR

This paper investigates the stability of solutions to elliptic M"obius difference equations, concluding that such sequences lack Hyers-Ulam stability under specified conditions.

Contribution

It establishes that solutions to elliptic M"obius difference equations do not exhibit Hyers-Ulam stability, clarifying the stability behavior of these complex dynamical systems.

Findings

Sequences have no Hyers-Ulam stability.

Stability depends on elliptic M"obius map properties.

Results contribute to complex dynamics and stability theory.

Abstract

The linear fractional map $ f(z) = \frac{az+ b}{cz + d} $ on the Riemann sphere with complex coefficients $ ad-bc \neq 0 $ is called M\"obius map. If $ f $ satisfies $ ad-bc=1 $ and $ -2<a+d<2 $, then $ f $ is called $\textit{elliptic}$ M\"obius map. Let $ \{ b_n \}_{n \in \mathbb{N}_0} $ be the solution of the elliptic M\"obius difference equation $ b_{n+1} = f(b_n) $ for every $ n \in \mathbb{N}_0 $. Then the sequence $ \{ b_n \}_{n \in \mathbb{N}_0} $ has no Hyers-Ulam stability.

Equations

$$|a_{n+1}-F(n,a_n)|\le \epsilon$$

$$b_{n+1}=F(n,b_n)$$

$$g(z)=\frac{az+b}{cz+d}$$

$$x=\frac{a-d\pm \sqrt{(a+d{)}^2-4}}{2c}.$$

$$h\circ g\circ h^{-1}(x)=\frac{x}{(c\alpha +d{)}^2}.$$

$$h^{\prime }(g(\alpha ))g^{\prime }(\alpha )=h^{\prime }(\alpha )g^{\prime }(\alpha )=k{h}^{\prime }(\alpha )$$

$$|c\alpha +d{|}^2$$

$$=c^2\alpha \bar{\alpha }+cd(\alpha +\bar{\alpha })+d^2$$

$$=c^2\left(-\frac{b}{c}\right)+cd\frac{a-d}{c}+d^2$$

$$=-bc+ad-d^2+d^2$$

$$=ad-bc$$

$$=1.$$

$$|h^{-1}(p)-h^{-1}(p^{\prime })|$$

$$=\left|\frac{(\bar{\alpha }p-\alpha )(p^{\prime }-1)-(\bar{\alpha }{p}^{\prime }-\alpha )(p-1)}{(p-1)(p^{\prime }-1)}\right|$$

$$=\left|\frac{(\alpha -\bar{\alpha })(p-p^{\prime })}{(p-1)(p^{\prime }-1)}\right|$$

$$=\left|\frac{\alpha -\bar{\alpha }}{(p-1)(p^{\prime }-1)}\right|\cdot |p-p^{\prime }|.$$

$$|h^{-1}(p)-g^{n_k}\circ h^{-1}(p_0)|$$

$$\le \left(\left|\frac{\alpha -\bar{\alpha }}{(p-1{)}^2}\right|+\delta \right)|p-p_{n_k}|$$

$$\alpha =\frac{a-d+\sqrt{(a+d{)}^2-4}}{2c},\beta =\frac{a-d-\sqrt{(a+d{)}^2-4}}{2c}$$

$$\alpha +\frac{d}{c}=\frac{a+d+\sqrt{(a+d{)}^2-4}}{2c},\beta +\frac{d}{c}=\frac{a+d-\sqrt{(a+d{)}^2-4}}{2c}$$

$$\left|\alpha +\frac{d}{c}\right|=\left|\beta +\frac{d}{c}\right|=\frac{1}{|c|}.$$

$$f(z)-\alpha$$

$$=\frac{az+b}{cz+d}-\frac{a\alpha +b}{c\alpha +d}=\frac{(ad-bc)(z-\alpha )}{(cz+d)(c\alpha +d)}=\frac{z-\alpha }{(cz+d)(c\alpha +d)}$$

$$f(z)-\beta =\frac{z-\beta }{(cz+d)(c\beta +d)}.$$

$$|a_{n+1}-f(a_n)|\le \epsilon$$

$$|c_k-d_k|=|c_0-d_k|=|c_0-d_0-n\epsilon |\ge -|c_0-d_0|+n\epsilon \ge (n-1)\epsilon$$

$$p(z)=\frac{\sqrt{3}z-2}{2z-\sqrt{3}},q(z)=\frac{-1}{z-\sqrt{3}},r(z)=\frac{-z-1}{z}.$$

Full text

1 Introduction

The first order difference equation is the solution of $b_{n+1}=F(n,b_{n})$ for $n\in\mathbb{N}_{0}$ with the initial point $b_{0}$. For the introductory method and examples see [3]. An interesting non-linear difference equation is the rational difference equation. For instance, Pielou logistic difference equation [7] or Beverton-Holt equation [2, 9] are first order rational difference equation as a model for population dynamics with constraint. These equations are understood as the iteration of a kind of Möbius transformation on the real line. In this paper, we investigate the Hyers-Ulam stability of another kind of Möbius transformation which does not appear in population dynamics and extend the result to the complex plane.

Hyers-Ulam stability raised from Ulam’s question [10] about the stability of approximate homomorphism between metric groups. The first answer to this question was given by Hyers [4] for Cauchy additive equation in Banach space. Later, the theory of Hyers-Ulam stability is developed in the area of functional equation and differential equation by many authors. The theory of Hyers-Ulam stability for difference equation appears in relatively recent decades and is mainly searched for linear difference equations, for example, see [5, 6, 8, 11] . Denote the set of natural numbers by $\mathbb{N}$ and denote the set $\mathbb{N}\cup\{\infty\}$ by $\mathbb{N}_{0}$. The set of real numbers and complex numbers by $\mathbb{R}$ and $\mathbb{C}$ respectively. Denote the unit circle by $\mathbb{S}^{1}$.

Suppose that the complex valued sequence $\{a_{n}\}_{n\in\mathbb{N}}$ satisfies the inequality

[TABLE]

for a $\varepsilon>0$ and for all $n\in\mathbb{N}_{0}$, where $|\cdot|$ is the absolute value of complex number. If there exists a sequence $\{b_{n}\}_{n\in\mathbb{N}}$ which satisfies that

[TABLE]

for each $n\in\mathbb{N}_{0}$ and $|a_{n}-b_{n}|\leq G(\varepsilon)$ for all $n\in\mathbb{N}_{0}$, where the positive number $G(\varepsilon)\rightarrow 0$ as $\varepsilon\rightarrow 0$. Then we say that the difference equation (1.1) has Hyers-Ulam stability.

Classification of Möbius transformation

Denote the Riemann sphere by $\hat{\mathbb{C}}$, which is the one point compactification of the complex plane, namely, $\mathbb{C}\cup\{\infty\}$. Similarly, we define the extended real line as $\mathbb{R}\cup\{\infty\}$ and denote it by $\hat{\mathbb{R}}$. Möbius transformation (or Möbius map) is the linear fractional map defined on $\hat{\mathbb{C}}$ as follows

[TABLE]

where $a,b,c$ and $d$ are complex numbers and $ad-bc\neq 0$. Define $g\left(-\frac{d}{c}\right)=\infty$ and $g(\infty)=\frac{a}{c}$. If $c=0$, then $g$ is the linear function. Thus we assume that $c\neq 0$ throughout this paper. The Möbius map which preserves $\hat{\mathbb{R}}$ is called the real Möbius map. A Möbius map is real if and only if the coefficients of the map $a,b,c$ and $d$ are real numbers.

The Möbius map has two fixed points counting with multiplicity. Denote these points by $\alpha$ and $\beta$. The real Möbius maps are classified to the three different cases using fixed points.

• •

If $\alpha$ and $\beta$ are real distinct numbers, the map is called real hyperbolic Möbius map,

• •

If $\alpha=\beta$, then the map is called real parabolic Möbius map, and

• •

If $\alpha$ and $\beta$ are two distinct non-real complex numbers, then the map is called real elliptic Möbius map.

Möbius map $x\mapsto\frac{ax+b}{cx+d}$ is the same as $x\mapsto\frac{pax+pb}{pcx+pd}$ for all numbers $p\neq 0$. Thus we may assume that $ad-bc=1$ when we choose $p=\sqrt{ad-bc}$. Moreover, Möbius map has the matrix representation \big{(}\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\big{)} under the condition $ad-bc=1$. Denote the matrix representation of the Möbius map $g$, by also $g$ and its trace by $\mathrm{tr}(g)$, which means $a+d$. In the complex analysis or hyperbolic geometry, Möbius maps with complex coefficients can be classified similarly with different method. For instance, see [1]. Möbius transformation in (1.2) (with real or complex coefficients) for $ad-bc=1$ is classified as follows

• •

If $\mathrm{tr}(g)\in\mathbb{R}\setminus[-2,2]$, then $g$ is called hyperbolic,

• •

If $\mathrm{tr}(g)=\pm 2$, then $g$ is called parabolic,

• •

If $\mathrm{tr}(g)\in(-2,2)$, then $g$ is called elliptic and

• •

If $\mathrm{tr}(g)\in\mathbb{C}\setminus\mathbb{R}$, then $g$ is called purely loxodromic.

The real Möbius maps are also classified by the above notions. In this paper, we investigate Hyers-Ulam stability of elliptic Möbius transformations. Other cases would appear in the forthcoming papers.

2 No Hyers-Ulam stability with dense subset

In this section we prove non-stability in the sense of Hyers-Ulam, which is not only for the elliptic Möbius transformation but also for any function satisfying the assumption of the following Theorem 2.1.

Theorem 2.1**.**

Let $\{b_{n}\}_{n\in\mathbb{N}_{0}}$ be the sequence in $\mathbb{R}$ satisfying $b_{n+1}=F(b_{n})$ with a map $F$ for $n\in\mathbb{N}_{0}$. Suppose that there exists a dense subset $A$ of $\mathbb{R}$ such that if $b_{0}\in A$, then the sequence $\{b_{n}\}_{n\in\mathbb{N}_{0}}$ is dense in $\mathbb{R}$. Suppose also that $\{b_{n}\}_{n\in\mathbb{N}_{0}}$ has no periodic point. Then the sequence $\{b_{n}\}_{n\in\mathbb{N}_{0}}$ has no Hyers-Ulam stability.

Proof.

For any $a_{0}\in\mathbb{R}$ and $\varepsilon>0$, choose the sequence $\{a_{n}\}_{n\in\mathbb{N}_{0}}$ as follows

$a_{0}\in\mathbb{R}$ is arbitrary, 2. 2.

$a_{1}$ satisfies that $|F(a_{0})-a_{1}|\leq\varepsilon$ and $a_{1}\in A$, that is, the sequence $\{F^{n}(a_{1})\}_{n\in\mathbb{N}_{0}}$ is dense in $\mathbb{R}$.

Let $n_{k}$ for $k\geq 1$ be the positive numbers $n_{1}<n_{2}<\cdots<n_{k}<\cdots$ such that $F^{n_{k}}(a_{1})$ are points in the ball of which center is $a_{1}$ and diameter is $\varepsilon$.

$a_{n+1}=F^{n}(a_{1})$ for $n=0,1,2,\ldots,n_{1}-1$, 2. 4.

$a_{n_{1}+1}=a_{1}$  and  $a_{k}=a_{k+n_{1}+1}$ for every $k\in\mathbb{N}$.

Then the sequence $\{a_{n}\}_{n\in\mathbb{N}_{0}}$ satisfies that $|a_{n+1}-F(a_{n})|\leq\varepsilon$ for all $n\in\mathbb{N}_{0}$. Moreover, since the sequence $\{a_{n}\}_{n\in\mathbb{N}}$ is periodic, $\{a_{n}\}_{n\in\mathbb{N}_{0}}$ is the finite set and it is bounded. However, the fact that the sequence $\{b_{n}\}_{n\in\mathbb{N}_{0}}$ is dense in $\mathbb{R}$ implies that $|a_{n}-b_{n}|$ is unbounded for $n\in\mathbb{N}$. Hence, the sequence $\{b_{n}\}_{n\in\mathbb{N}_{0}}$ does not have Hyers-Ulam stability. ∎

Remark 2.2**.**

Theorem 2.1 can be generalized to any metric space only if the definition of Hyers-Ulam stability is modified suitably. For instance, if $F$ is the map from the metric space $X$ to itself and $|\cdot|$ is changed to the distance $\mathrm{dist}(\cdot\,,\cdot)$ from the metric on $X$, then we can define Hyers-Ulam stability on the metric space and Theorem 2.1 is applied to it. For example, the unit circle $\mathbb{S}^{1}$ is the metric space of which distance between two points defined from the minimal arc length connecting these two points. Then Hyers-Ulam stability on $\mathbb{S}^{1}$ can be defined.

3 Real elliptic Möbius transformation

Lemma 3.1**.**

Let $g(x)=\frac{ax+b}{cx+d}$ be the linear fractional map where $a,b,c$ and $d$ are real numbers, $c\neq 0$ and $ad-bc=1$. Then $g$ has fixed points which are non-real complex numbers if and only if $g$ is real elliptic Möbius map, that is, $-2<a+d<2$.

Proof.

The equation $g(x)=x$ implies that

[TABLE]

Then the fixed points are non-real complex numbers if and only if $-2<a+d<2$. ∎

Lemma 3.2**.**

Let $g$ be the map defined on $\hat{\mathbb{C}}$ in Lemma 3.1 and two complex numbers $\alpha$ and its complex conjugate $\bar{\alpha}$ be the fixed points of $g$. Let $h$ be the map defined as $h(x)=\frac{x-\alpha}{x-\bar{\alpha}}$. If $g$ is the elliptic Möbius map, that is, $-2<a+d<2$, then

[TABLE]

for $x\in\hat{\mathbb{C}}$. Moreover, $|g^{\prime}(\alpha)|=|c\alpha+d|=1$.

Proof.

The map $h\circ g\circ h^{-1}$ has the fixed points [math] and $\infty$. Since both $g$ and $h$ are linear fractional maps, so is $h\circ g\circ h^{-1}$. Then $h\circ g\circ h^{-1}(x)=kx$ for some $k\in\mathbb{C}$. The equation $h\circ g(x)=kh(x)$ implies that $h^{\prime}(g(x))g^{\prime}(x)=kh^{\prime}(x)$. Thus

[TABLE]

Then $k=g^{\prime}(\alpha)=\frac{1}{(c\alpha+d)^{2}}$. Moreover, $|g^{\prime}(\alpha)|=1$ if and only if $|c\alpha+d|=1$. Recall that both $\alpha$ and $\bar{\alpha}$ are roots of the equation, $cx^{2}-(a-d)x-b=0$. Then

[TABLE]

Hence, $|g^{\prime}(\alpha)|=\frac{1}{|c\alpha+d|^{2}}=1$. ∎

By Lemma 3.2, the map $h\circ g\circ h^{-1}$ is a rotation on $\mathbb{S}^{1}$. Since $h^{-1}$ is bijective from $\mathbb{S}^{1}\setminus\{1\}$ to $\mathbb{R}$, $x$ is a periodic point under $h\circ g\circ h^{-1}$ in $\mathbb{S}^{1}\setminus\{1\}$ with period $p$ if and only if $h^{-1}(x)$ is periodic in $\mathbb{R}$ with the same period. Recall that $h^{-1}(1)=\infty$. Thus when we investigate Hyers-Ulam stability of the sequence $\{b_{n}\}_{n\in\mathbb{N}_{0}}$ as the solution of the elliptic linear fractional map $g$, we have to choose carefully the initial point $b_{0}\in\mathbb{R}$ satisfying $g^{k}(b_{0})\neq\infty$ for all $k\in\mathbb{N}$.

Proposition 3.3**.**

Let $g$ be the elliptic linear fractional map defined in Lemma 3.1 on $\hat{\mathbb{R}}$. Suppose that there exists $x\in\mathbb{R}$ such that $g^{k}(x)\neq x$ for all $k\in\mathbb{N}$. Then the sequence $\{g^{k}(x)\}_{k\in\mathbb{N}}$ is dense in $\mathbb{R}$ where $x\in\mathbb{R}\setminus\{g^{-k}(\infty)\}_{k\in\mathbb{N}}$.

Proof.

Lemma 3.2 implies that $h\circ g\circ h^{-1}(x)=e^{i\theta}x$ for some $\theta\in\mathbb{R}$. If $\theta$ is a rational number $\frac{q}{p}$, then $h\circ g^{p}\circ h^{-1}(x)=x$ for all $x\in\mathbb{C}$. Thus $g^{p}(x)=x$ for all $x\in\mathbb{C}$. Then $\theta$ is an irrational number. Since $x\mapsto e^{i\theta}x$ is an irrational rotation on $\mathbb{S}^{1}$, the sequence $\{e^{ik\theta}x\}_{k\in\mathbb{N}}$ is dense in $\mathbb{S}^{1}$ for every $x\in\mathbb{S}^{1}$. Moreover, when $x=1$ is chosen, the set $\mathbb{S}^{1}\setminus\{e^{-ik\theta}\}_{k\in\mathbb{N}_{0}}$ is also a dense subset of $\mathbb{S}^{1}$.

The direct calculation implies that $h^{-1}(x)=\frac{\bar{\alpha}x-\alpha}{x-1}$ and then $h^{-1}(1)=\infty$. Choose two points $p$ and $p^{\prime}$ close enough to each other in $\mathbb{S}^{1}\setminus\{e^{-ik\theta}\}_{k\in\mathbb{N}_{0}}$. Then

[TABLE]

We choose the sequence $\{p_{n_{k}}\}_{k\in\mathbb{N}_{0}}$ for some $p_{0}\in\mathbb{S}^{1}\setminus\{e^{-ik\theta}\}_{k\in\mathbb{N}_{0}}$ which satisfies that $h\circ g^{n_{k}}\circ h^{-1}(p_{0})=p_{n_{k}}$ and $p_{n_{k}}\rightarrow p$ as $k\rightarrow\infty$ for different numbers $n_{1}<n_{2}<\cdots<n_{k}<\cdots$. Since $h^{-1}$ is a bijection from $\mathbb{S}^{1}\setminus\{1\}$ to $\mathbb{R}$, by the equation (3) we obtain that

[TABLE]

for some $\delta>0$. Then any point $h^{-1}(p)$ in $\mathbb{R}$ is an accumulation point in the sequence $\{g^{k}(x)\}_{k\in\mathbb{N}_{0}}$ where x\in h^{-1}\big{(}\mathbb{S}^{1}\setminus\{e^{-ik\theta}\}_{k\in\mathbb{N}_{0}}\big{)}, which is a dense subset of $\mathbb{R}$. ∎

Theorem 2.1 and Proposition 3.3 imply that the real elliptic linear fractional map does not have Hyers-Ulam stability.

Corollary 3.4**.**

Let $g(x)=\frac{ax+b}{cx+d}$ be the linear fractional map on $\hat{\mathbb{R}}$ where $a,b,c$ and $d$ are real numbers, $c\neq 0$ and $ad-bc=1$. Suppose that $-2<a+d<2$ and $h\circ g\circ h^{-1}$ is an irrational rotation on the unit circle where $h(x)=\frac{x-\alpha}{x-\bar{\alpha}}$. If the sequence $\{b_{n}\}_{n\in\mathbb{N}_{0}}$ in $\mathbb{R}$ is the solution of $b_{n+1}=g(b_{n})$ for $n\in\mathbb{N}_{0}$, then either $g^{k}(b_{0})=\infty$ for some $k\in\mathbb{N}$ or $\{b_{n}\}_{n\in\mathbb{N}_{0}}$ has no Hyers-Ulam stability .

4 Extension of non stability to complex plane

In this section, we extend no Hyers-Ulam stability of the real elliptic linear fractional map to the elliptic Möbius transformation with complex coefficients. Let $\ell$ be the straight line in the complex plane. Define the extended line as $\ell\cup\{\infty\}$ and denote it by $\hat{\ell}$. Interior of the circle $C$ in $\mathbb{C}$ means that the bounded region of the set $\mathbb{C}\setminus C$.

Lemma 4.1**.**

Let $f(z)=\frac{az+b}{cz+d}$ be the Möbius map with complex coefficients $a,b,c$ and $d$ for $c\neq 0$ and $ad-bc=1$. If $f$ is elliptic, that is, $-2<a+d<2$, then there exists the unique extended line which is invariant under $g$ in $\hat{\mathbb{C}}$.

Proof.

Let $\alpha$ and $\beta$ be the fixed points of $f$. In particular, the fixed points of $g$ are as follows

[TABLE]

Denote the straight line, $\{z\in\mathbb{C}:|z-\alpha|=|z-\beta|\}$ by $\ell$ in $\mathbb{C}$ and denote the extended line $\ell\cup\{\infty\}$ by $\hat{\ell}$. We prove that $\hat{\ell}$ is the unique invariant extended line under $f$.

Claim : The points, $-\frac{d}{c}$ and $\frac{a}{c}$ are in $\hat{\ell}$.

[TABLE]

The fact that $a+d$ is a real number and $\sqrt{(a+d)^{2}-4}$ is a purely imaginary number implies that

[TABLE]

Then $-\frac{d}{c}$ is in $\hat{\ell}$. By similar calculation, $\frac{a}{c}$ is also in $\hat{\ell}$. The proof the claim is complete.

For the invariance of $\hat{\ell}$, it suffice to show that if $z\in{\ell}\setminus\{-\frac{d}{c}\}$, then $f(z)\in\ell$. For any $z\in{\ell}\setminus\{-\frac{d}{c}\}$, we have that

[TABLE]

By similar calculation, we obtain that

[TABLE]

The equation (4.1) in the claim implies that $|c\alpha+d|=|c\beta+d|$. The definition of $\ell$ implies that $cz+d\neq 0$ for $z\in{\ell}\setminus\{-\frac{d}{c}\}$ and $|z-\alpha|=|z-\beta|$. The equation $|f(z)-\alpha|=|f(z)-\beta|$ holds by comparing the equation (4) with (4.3). Hence, by this result and the above claim, the extended line $\hat{\ell}$ is invariant under $f$. There is the unique straight line which connects $-\frac{d}{c}$ and $\frac{a}{c}$ in $\mathbb{C}$. Hence, $\hat{\ell}$ is the unique invariant extended line in $\hat{\mathbb{C}}$ under $f$. ∎

Remark 4.2**.**

Define the map $h$ as $h(z)=\frac{z-\beta}{z-\alpha}$. Then by the straightforward calculation we obtain $h\circ f\circ h^{-1}(z)=f^{\prime}(\alpha)z$ and $|f^{\prime}(\alpha)|=1$. Let $C_{r}$ be the circle of radius $r>0$ of which center is the origin in $\mathbb{C}$. Observe that a fixed point of $f$ is contained in the interior of $h^{-1}(C_{r})$ for all $r>0$. If $h\circ g\circ h^{-1}$ is an irrational rotation in $\mathbb{C}$, then every concentric circles $C_{r}$ are invariant under $h\circ f\circ h^{-1}$. Then $h^{-1}(C_{r})$ is an invariant circle under $f$ for every $r>0$. However, since $h(\infty)=1$, the unique invariant extended line under $f$ is $h^{-1}(\mathbb{S}^{1})$. Moreover, $h^{-1}(C_{1/r})$ is the reflected image of $h^{-1}(C_{r})$ to the invariant extended line.

Due to the existence of the extended line which is invariant under $g$, no Hyers-Ulam stability of the sequence $\{g^{n}(z)\}_{n\in\mathbb{N}_{0}}$ of the real elliptic linear fractional map is extendible to that of the elliptic Möbius transformation.

Proposition 4.3**.**

Let $f$ be the elliptic Möbius transformation on $\hat{\mathbb{C}}$ and $\hat{\ell}$ be the extended line invariant under $f$. Suppose that there exists $z\in\hat{\ell}$ such that $f^{k}(z)\neq z$ for all $k\in\mathbb{N}$. Then the sequence $\{f^{k}(z)\}_{k\in\mathbb{N}}$ is dense in $\ell=\hat{\ell}\setminus\{\infty\}$ where $z\in\ell\setminus\{f^{-k}(\infty)\}_{k\in\mathbb{N}}$.

Proof.

Lemma 4.1 implies that there exists the invariant extended line under $f$. In the proof of Proposition 3.3, replace $\bar{\alpha}$ by $\beta$ and apply the proof of Proposition 3.3. Then the similar calculation completes the proof. ∎

Then Proposition 4.3 and Theorem 2.1 implies that no Hyers-Ulam stability of $\{f^{n}(z)\}_{n\in\mathbb{N}_{0}}$ where $f$ is the elliptic Möbius map for $z\in\mathbb{C}$.

Corollary 4.4**.**

Let $f(z)=\frac{az+b}{cz+d}$ be the Möbius map on $\hat{\mathbb{C}}$ where $c\neq 0$ and $ad-bc=1$. If $f$ is elliptic, that is, $-2<a+d<2$, then the sequence $\{b_{n}\}_{n\in\mathbb{N}_{0}}$ satisfying $b_{n+1}=f(b_{n})$ for $n\in\mathbb{N}_{0}$ has no Hyers-Ulam stability on $\mathbb{C}$.

Proof.

The sequence $\{a_{n}\}_{n\in\mathbb{N}_{0}}$ is in the invariant extended line $\hat{\ell}$ under $f$ satisfying

[TABLE]

for $n\in\mathbb{N}_{0}$. Choose $\{a_{n}\}_{n\in\mathbb{N}_{0}}$ is the finite sequence which satisfies the properties in the proof of Theorem 2.1 in $\ell$. Then the similar proof in Section 3 implies that any sequence $\{b_{n}\}_{n\in\mathbb{N}_{0}}$ satisfying $b_{n+1}=f(b_{n})$ for every $n\in\mathbb{N}_{0}$ in $\ell$ has no Hyers-Ulam stability.

Let $h$ be the map $h(z)=\frac{z-\alpha}{z-\beta}$. Observe that $h\circ f\circ h^{-1}$ is an irrational rotation on $\mathbb{C}$. If $\{b_{n}\}_{n\in\mathbb{N}_{0}}$ is contained in $\mathbb{C}\setminus\hat{\ell}$, then $\{b_{n}\}_{n\in\mathbb{N}_{0}}$ is contained in the circle disjoint from $\ell$ and this circle is $h^{-1}(C_{r})$ for some $r>0$. Observe that if $r=0$, then $C_{r}=\{\alpha\}$ and if $r=1$, then $C_{r}=\hat{\ell}$. Since $h\circ f\circ h^{-1}$ is an irrational rotation, $\{f^{n}(h^{-1}(x))\}_{n\in\mathbb{N}_{0}}$ is the dense subset of $h^{-1}(C_{r})$. We may assume that $0<r<1$ and the fixed point $\alpha$ is contained in the interior of $C_{r}$ by Remark 4.2. Denote the (minimal) distance between the point $\alpha$ and the line ${\ell}$ by $L$. Then the density of $\{b_{n}\}_{n\in\mathbb{N}_{0}}$ on the circle $C_{r}$ and the finiteness of $\{a_{n}\}_{n\in\mathbb{N}_{0}}$ imply that $|a_{n}-b_{n}|\geq L$ for infinitely many $n\in\mathbb{N}$ for all $\varepsilon<L$. Hence, $\{b_{n}\}_{n\in\mathbb{N}_{0}}$ has no Hyers-Ulam stability. ∎

5 Non stability of periodic sequence

Let the sequence $\{b_{n}\}_{n\in\mathbb{N}_{0}}$ satisfying $b_{n+p}=b_{n}$ for every $n\in\mathbb{N}_{0}$ for some $p\in\mathbb{N}$ be periodic sequence. The least positive number $p$ satisfying the above equation is called the peroid of sequence. If $p=1$, then it is called constant sequence.

Lemma 5.1**.**

Let the sequence $\{b_{n}\}_{n\in\mathbb{N}_{0}}$ in $\mathbb{R}$ be a periodic sequence with period $p$. Then $\{b_{n}\}_{n\in\mathbb{N}_{0}}$ has no Hyers-Ulam stability.

Proof.

Any periodic sequence has constant subsequence. For example, let $\{c_{n}\}_{n\in\mathbb{N}_{0}}$ be the sequence satisfying $c_{n}=b_{pn}$ for every $n\in\mathbb{N}_{0}$. Thus $\{c_{n}\}_{n\in\mathbb{N}_{0}}$ is the constant sequence $\{c_{0}\}_{n\in\mathbb{N}_{0}}$. It suffice to show that the constant sequence has no Hyers-Ulam stability. For any small enough $\varepsilon>0$, define the sequence $\{d_{n}\}_{n\in\mathbb{N}_{0}}$ as follows

$d_{0}$ is arbitrary and 2. 2.

$d_{n}=d_{0}+n\varepsilon$ for $n\in\mathbb{N}$

Then $|d_{n+1}-d_{n}|\leq\varepsilon$ for all $n\in\mathbb{N}_{0}$. However, for any constant sequence $\{c_{0}\}_{n\in\mathbb{N}_{0}}$ satisfying $|c_{0}-d_{0}|\leq\varepsilon$ such that

[TABLE]

for all $n\geq 2$. Since $|c_{k}-d_{k}|$ is unbounded for $n\in\mathbb{N}$, the constant sequence $\{c_{n}\}_{n\in\mathbb{N}_{0}}$ has no Hyers-Ulam stability. Hence, the periodic sequence $\{b_{n}\}_{n\in\mathbb{N}_{0}}$ has no Hyers-Ulam stability either. ∎

Example 5.2**.**

There are non-linear Möbius transformations, of which finitely many composition is the identity map. For example, see the following maps

[TABLE]

Thus $p\circ p(z)=p^{2}(z)=z$ for all $z\in\mathbb{C}$ and the trace of $p$ is that $\mathrm{tr}(p)=\sqrt{3}+(-\sqrt{3})=0$. The map $q$ satisfies that $q^{3}=p$ by the direct calculation and then $q^{6}(z)=z$ for all $z\in\mathbb{C}$. The trace of $q$ is that $\mathrm{tr}(q)=0+(-\sqrt{3})=-\sqrt{3}$. Finally, $r^{3}(z)=z$ for all $z\in\mathbb{C}$ and $\mathrm{tr}(r)=-1+0=-1$. All of the traces of $p$, $q$ and $r$ are between $-2$ and $2$.

Remark 4.2 implies that $h\circ g\circ h^{-1}(x)=e^{i\theta}x$ for every elliptic Möbius map $g$. If $\theta=\frac{q}{p}$, then $g^{p}(x)=x$ for all $x\in\mathbb{C}$. Thus the sequence $\{g^{n}(x)\}_{n\in\mathbb{N}_{0}}$ is periodic. Then Corollary 3.4 and Lemma 5.1 implies the following theorem.

Theorem 5.3**.**

Let $g(x)=\frac{ax+b}{cx+d}$ be the linear fractional map on $\hat{\mathbb{R}}$ for $c\neq 0$, $ad-bc=1$. Supposet that $g$ is the elliptic linear fractional map, that is, $-2<a+d<2$. Then the sequence $\{b_{n}\}_{n\in\mathbb{N}_{0}}$ in $\mathbb{R}$ satisfying $b_{n+1}=g(b_{n})$ for $n\in\mathbb{N}_{0}$ either satisfies that $g^{k}(b_{0})=\infty$ for some $k\in\mathbb{N}$ or it has no Hyers-Ulam stabiliy.

If the sequence $\{f^{n}(z)\}_{n\in\mathbb{N}_{0}}$ is periodic, then Corollary 4.4 and Lemma 5.1 implies the following theorem.

Theorem 5.4**.**

Let $f(z)=\frac{az+b}{cz+d}$ be the Möbius map on $\hat{\mathbb{C}}$ for $ad-bc=1$, $c\neq 0$. Suppose that $f$ is the elliptic Möbius map, that is, $-2<a+d<2$. Then the sequence $\{b_{n}\}_{n\in\mathbb{N}_{0}}$ in $\mathbb{C}$ satisfying $b_{n+1}=f(b_{n})$ for $n\in\mathbb{N}_{0}$ either satisfies that $g^{k}(b_{0})=\infty$ for some $k\in\mathbb{N}$ or it has no Hyers-Ulam stabiliy.