Hyers-Ulam stability of elliptic Möbius difference equation

Abstract: The linear fractional map fðzÞ 1⁄4 azþb czþd on the Riemann sphere with complex coefficients ad bc 0 is called Möbius map. If f satisfies ad bc 1⁄4 1 and 2<aþ d < 2, then f is called ellipticMöbiusmap. Let bn f gn2N0 be the solution of the elliptic Möbius difference equation bnþ1 1⁄4 fðbnÞ for every n 2 N0. We show that the sequence bn f gn2N0 on the complex plane as well as on the real numbers has no Hyers-Ulam stability by conjugation method.


Introduction
The first order difference equation is defined as the solution of b nþ1 ¼ Fðn; b n Þ for n 2 N 0 with the initial point b 0 . An interesting non-linear difference equation is the rational difference equation. For instance, Pielou logistic difference equation (Pielou, 1974) or Beverton-Holt equation (Bohner & Warth, 2007;Sen, 2008) are the 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. For the introduction and examples of difference equation defined on the real line, see (Elaydi, 2005).
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 (Ulam, 1960) about the stability of approximate homomorphism between metric groups. The first answer to this question was given by Hyers (Hyers, 1941) 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 (Jung, 2015;Jung & Nam, ABOUT THE AUTHOR Young Woo Nam is an invited professor of Mathematics Section, College of Science and Technology, Hongik University at Sejong, Korea. His fields of interest are renormalization in dynamical systems, Hyers-Ulam stability in difference equation.

PUBLIC INTEREST STATEMENT
Difference equation is a field of mathematics which may describe the discrete dynamical systems. It has been successfully used to describing some phenomenon, for instance, population dynamics. In this article, a certain type of difference equations has no control of errors even though the amount of error of each term is arbitrarily small. The equation in the article is the elliptic linear fractional map and it is a kind of rotation where the map is defined on the set of complex numbers. 2016; Popa, 2015;Xu & Brzdek, 2015). Denote the set of natural numbers by N and denote the set N[ 1 f g by N 0 . The set of real numbers and complex numbers by R and C, respectively. Denote the unit circle by S 1 .
Suppose that the complex valued sequence a n f g n2N satisfies the inequality a nþ1 À Fðn; a n Þ j j ε for a ε > 0 and for all n 2 N 0 , where Á j j is the absolute value of complex number. If there exists a sequence b n f g n2N which satisfies that b nþ1 ¼ Fðn; b n Þ (1:1) for each n 2 N 0 and a n À b n j j GðεÞ for all n 2 N 0 , where the positive number GðεÞ ! 0 as ε ! 0. Then we say that the difference Equation (1.1) has Hyers-Ulam stability.

Classification of Möbius transformation
Denote the Riemann sphere byĈ, which is the one point compactification of the complex plane, namely, C[ 1 f g . Similarly, we define the extended real line as R[ 1 f g and denote it byR. Möbius transformation (or Möbius map) is the linear fractional map defined onĈ as follows where a; b; c and d are complex numbers and ad À bc Þ 0. Define g À d c À Á ¼ 1 and gð1Þ ¼ a c . If c ¼ 0, then g is the linear function. Thus we assume that c Þ 0 throughout this article. The Möbius map which preservesR 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 α and β. The real Möbius maps are classified to the three different cases using fixed points.
• If α and β are real distinct numbers, the map is called real hyperbolic Möbius map, • If α ¼ β, then the map is called real parabolic Möbius map, and • If α and β are two distinct non-real complex numbers, then the map is called real elliptic Möbius map.
Möbius map x7 ! axþb cxþd is the same as x7 ! paxþpb pcxþpd for all numbers p Þ 0. Thus we may assume that ad À bc ¼ 1 when we choose p ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ad À bc p . Moreover, Möbius map has the matrix representation a b c d under the condition ad À bc ¼ 1. Denote the matrix representation of the Möbius map g, by also g and its trace by trðgÞ, which means a þ d. In the complex analysis or hyperbolic geometry, Möbius maps with complex coefficients can be classified similarly with different method. For instance, see (Beardon, 1983). Möbius transformation in (1.2) (with real or complex coefficients) for ad À bc ¼ 1 is classified as follows • If trðgÞ 2 Rn½À2; 2, then g is called hyperbolic, • If trðgÞ ¼ AE2, then g is called parabolic, • If trðgÞ 2 ðÀ2; 2Þ, then g is called elliptic and • If trðgÞ 2 CnR, then g is called purely loxodromic.
The real Möbius maps are also classified by the above notions. In this article, we investigate Hyers-Ulam stability of elliptic Möbius transformations. Other cases would appear in the forthcoming articles.

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.
Suppose also that b n f g n2N 0 has no periodic point. Then the sequence b n f g n2N 0 has no Hyers-Ulam stability.
Proof. For any a 0 2 R and ε > 0, choose the sequence a n f g n2N 0 as follows (1) a 0 2 R is arbitrary, (2) a 1 satisfies that Fða 0 Þ À a 1 j j ε and a 1 2 A, that is, the sequence F n ða 1 Þ f g n2N 0 is dense in R.
Let n k for k ! 1 be the positive numbers n 1 < n 2 < Á Á Á < n k < Á Á Á such that F n k ða 1 Þ are points in the ball of which center is a 1 and diameter is ε.
Then the sequence a n f g n2N 0 satisfies that a nþ1 À Fða n Þ j j ε for all n 2 N 0 . Moreover, since the sequence a n f g n2N is periodic, a n f g n2N 0 is the finite set and it is bounded. However, the fact that the sequence b n f g n2N 0 is dense in R implies that a n À b n j jis unbounded for n 2 N. Hence, the sequence b n f g n2N 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 Á j j is changed to the distance distðÁ; ÁÞ 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 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 S 1 can be defined.

Real elliptic Möbius transformation
Lemma 3.1 Let gðxÞ ¼ axþb cxþd be the linear fractional map where a; b; c and d are real numbers, cÞ0 and ad À bc ¼ 1. Then g has fixed points which are non-real complex numbers if and only if g is real elliptic Möbius map, that is, Then the fixed points are non-real complex numbers if and only if À 2 < a þ d < 2. • Lemma 3.2 Let g be the map defined onĈ in Lemma 3.1 and two complex numbers α and its complex conjugate α be the fixed points of g. Let h be the map defined as hðxÞ ¼ xÀα xÀ α . If g is the elliptic Möbius map, that is, À 2 < a þ d < 2, then h g h À1 ðxÞ ¼ x ðcα þ dÞ 2 : for x 2Ĉ. Moreover, g 0 ðαÞ ¼ j jcα þ d j j ¼ 1.
Proof. The map h g h À1 has the fixed points 0 and 1.
Hence; g 0 ðαÞ j j¼ 1 jcα þ dj 2 ¼ 1: By Lemma 3.2, the map h g h À1 is a rotation on S 1 . Since h À1 is bijective from S 1 n 1 f g to R, x is a periodic point under h g h À1 in S 1 n 1 f g with period p if and only if h À1 ðxÞ is periodic in R with the same period. Recall that h À1 ð1Þ ¼ 1. Thus when we investigate Hyers-Ulam stability of the sequence b n f g n2N 0 as the solution of the elliptic linear fractional map g, we have to choose carefully the initial point b 0 2 R satisfying g k ðb 0 ÞÞ1 for all k 2 N.
Proposition 3.3. Let g be the elliptic linear fractional map defined in Lemma 3.1 onR. Suppose that there exists x 2 R such that g k ðxÞÞx for all k 2 N. Then the sequence g k ðxÞ È É k2N is dense in R where x 2 Rn g Àk ð1Þ È É k2N .
Proof. Lemma 3.2 implies that h g h À1 ðxÞ ¼ e iθ x for some θ 2 R. If 2πθ is a rational number q p , then h g p h À1 ðxÞ ¼ x for all x 2 C. Thus g p ðxÞ ¼ x for all x 2 C. Then 2πθ is an irrational number.
Since x7 !e iθ x is an irrational rotation on S 1 , the sequence e ikθ x È É k2N is dense in S 1 for every x 2 S 1 . Moreover, when x ¼ 1 is chosen, the set S 1 n e Àikθ È É k2N 0 is also a dense subset of S 1 .
The direct calculation implies that h À1 ðxÞ ¼ αxÀα xÀ1 and then h À1 ð1Þ ¼ 1. Choose two points p and p 0 close enough to each other in S 1 n e Àikθ È É k2N 0 . Then We choose the sequence p n k È É k2N 0 for some p 0 2 S 1 n e Àikθ È É k2N 0 which satisfies that h g n k h À1 ðp 0 Þ ¼ p n k and p n k ! p as k ! 1 for different numbers n 1 < n 2 < Á Á Á < n k < Á Á Á . Since h À1 is a bijection from S 1 n 1 f g to R, by the Equation (3.1) we obtain that h À1 ðpÞ À g n k h À1 ðp 0 Þ ¼ j jh À1 ðpÞ À h À1 ðp n k Þ α À α ðp À 1Þ 2 þ δ ! p À p n k for some δ > 0. Then any point h À1 ðpÞ in R is an accumulation point in the sequence g k ðxÞ Þ, which is a dense subset of 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 gðxÞ ¼ axþb cxþd be the linear fractional map onR where a; b; c and d are real numbers, cÞ0 and ad À bc ¼ 1. Suppose that À 2 < a þ d < 2 and h g h À1 is an irrational rotation on the unit circle where hðxÞ ¼ xÀα xÀ α . If the sequence b n f g n2N 0 in R is the solution of b nþ1 ¼ gðb n Þ for n 2 N 0 , then either g k ðb 0 Þ ¼ 1 for some k 2 N or b n f g n2N 0 has no Hyers-Ulam stability .
Example 3.5 The linear fractional map g is as follows Since number 1 ffiffiffiffi 13 p À 3 ffiffiffiffi 13 p ¼ À 2 ffiffiffiffi 13 p is between À 2 and 2, the map g is a real elliptic Möbius map. x. If we denote À11þ4 ffiffiffiffi 13 p i 13 by e iθ , then θ ¼ cos À1 11 13 À Á . Then 2πθ is an irrational number. Then for any x 2 R, either g n ðxÞ ¼ 1 for some n 2 N or the sequence g k ðxÞ È É k2N is dense in R by Proposition 3.3

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 , be the straight line in the complex plane. Define the extended line as ,[ 1 f g and denote it by,. Interior of the circle C in C means that the bounded region of the set CnC.
Lemma 4.1. Let f ðzÞ ¼ azþb czþd be the Möbius map with complex coefficients a; b; c and d for c Þ 0 and ad À bc ¼ 1. If f is elliptic, that is, À 2 < a þ d < 2, then there exists the unique extended line which is invariant under g inĈ.
Proof. Let α and β be the fixed points of f . In particular, the fixed points of g are as follows Denote the straight line, z 2 C : z À α j j¼ z À β j j f g by , in C and denote the extended line ,[ 1 f g bŷ ,. We prove that, is the unique invariant extended line under f .

Claim:
The points, À d c and a c are in,. Nam, Cogent Mathematics & Statistics (2018), 5: 1492338 https://doi.org/10. 1080/25742558.2018.1492338 The fact that a þ d is a real number and ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ða þ dÞ 2 À 4 q is a purely imaginary number implies that (4:1) Then À d c is in,. By similar calculation, a c is also in,. The proof the claim is complete.
For the invariance of,, it suffice to show that if z 2 ,n À d c È É , then f ðzÞ 2 ,. For any z 2 ,n À d c È É , we have that By similar calculation, we obtain that The Equation ( Corollary 4.4. Let f ðzÞ ¼ azþb czþd be the Möbius map onĈ where cÞ0 and ad À bc ¼ 1. If f is elliptic, that is, À 2 < a þ d < 2, then the sequence b n f g n2N 0 satisfying b nþ1 ¼ f ðb n Þ for n 2 N 0 has no Hyers- Ulam stability on C.
Proof. The sequence a n f g n2N 0 is in the invariant extended line, under f satisfying a nþ1 À f ða n Þ j j ε Theorem 5.3. Let gðxÞ ¼ axþb cxþd be the linear fractional map onR for c Þ 0, ad À bc ¼ 1. Supposet that g is the elliptic linear fractional map, that is, À 2 < a þ d < 2. Then the sequence b n f g n2N 0 in R satisfying b nþ1 ¼ gðb n Þ for n 2 N 0 either satisfies that g k ðb 0 Þ ¼ 1 for some k 2 N or it has no Hyers-Ulam stabiliy.
If the sequence f n ðzÞ f g n2N 0 is periodic, then Corollary 4.4 and Lemma 5.1 implies the following theorem.
Theorem 5.4. Let f ðzÞ ¼ azþb czþd be the Möbius map onĈ for ad À bc ¼ 1, c Þ 0. Suppose that f is the elliptic Möbius map, that is, À 2 < a þ d < 2. Then the sequence b n f g n2N 0 in C satisfying b nþ1 ¼ f ðb n Þ for n 2 N 0 either satisfies that g k ðb 0 Þ ¼ 1 for some k 2 N or it has no Hyers-Ulam stabiliy.

Conclusion and further research
In this paper, we show that the difference equation from the linear fractional map of elliptic type has no Hyers-Ulam stability. Using conjugation, this type of linear fractional map is actually a kind of rotation around the fixed points. The irrational rotation has dense orbits on every invariant circles and the rational rotation has all points which is periodic. Any of both maps does not have Hyers-Ulam stability. In the future, we investigate Hyers-Ulam stability of the difference equation from the linear fractional map of other types-parabolic, hyperbolic and loxodromic equations.

Funding
The author received no direct funding for this research.