In this paper we introduce expansiveiterated function systems, (IFS) on a compact metric space then various shadowing properties and their equivalence are considered for expans- iveIFS

Vol. 19 (2018), No. 1, pp. 413–422 DOI: 10.18514/MMN.2018.1623

VARIOUS SHADOWING PROPERTIES AND THEIR EQUIVALENT FOR EXPANSIVE ITERATED FUNCTION SYSTEMS

MEHDI FATEHI NIA Received 08 April, 2015

Abstract. In this paper we introduce expansiveiterated function systems, (IFS) on a compact metric space then various shadowing properties and their equivalence are considered for expans- iveIFS.

2010Mathematics Subject Classification: 37C50; 37C15

Keywords: eExpansive IFS, pseudo orbit, shadowing, continuous shadowing, limit shadowing, Lipschitz shadowing

1. INTRODUCTION

The notion of shadowing plays an important role in dynamical systems, specially;

in stability theory [1,11,12]. Various shadowing properties for expansive maps and their equivalence have been studied by Lee and Sakai [10,14]. More precisely, they prove the following theorems:

Theorem 1([14]). Letf be an expansive homeomorphism on a compact metric space.XId /. Then the following conditions are mutually equivalent:

.a/ f has the shadowing property,

.b/ f has the continuous shadowing property,

.c/ there is a compatible metricD forX such that f has the Lipschitz shadowing property with respect toD,

.d / f has the limit shadowing property,

.e/ there is a compatible metric D for X such that f has the strong shadowing property with respect toD.

Theorem 2([10]). Letf be a positively expansive map on a compact metrizable spaceX. Then the following conditions are mutually equivalent:

.a/ f is an open map,

.b/ f has the shadowing property,

.c/there is a metric such thatf has the Lipschitz shadowing property, .d /there is a metric such thatf has thes-limit shadowing property, .e/there is a metric such thatf has the strong shadowing property.

c 2018 Miskolc University Press

In the other hand, iterated function systems(IFS), are used for the construction of deterministic fractals and have found numerous applications, in particular to image compression and image processing [2]. Important notions in dynamics like attractors, minimality, transitivity, and shadowing can be extended to IFS (see [3,4,7–9]). The authors defined the shadowing property for a parameterized iterated function system and prove that if a parameterized IFS is uniformly expanding ( or contracting), then it has the shadowing property[9].

In this paper we present an approach to shadowing property for iterated function sys- tems. At first, we introduceexpansive iterated function systemson a compact metric space. Then continuous shadowing, limit shadowing and Lipschitz shadowing prop- erties are defined for anIFS,F D fXIfj2gwhereis a nonempty finite set andfWX !X is homeomorphism, for all2. Theorems3and4are the main result of the present work. Actually in these theorems we prove that the limit shad- owing property, the Lipschitz shadowing property are all equivalent to the shadowing property for expansive IFS on a compact metric space. The method is essentially the same as that used in [10,13,14]. Finally, we introduce the strong expansiveIFS and show that for a strong expansiveIFSthe continuous shadowing property and the shadowing property are equivalent.

2. PRELIMINARIES

In this section, we give some definitions and notations as well as some preliminary results that are needed in the sequel. Let.X; d /be a complete metric space. Let us recall that an Iterated Function System(IFS)

F D fXIfj2gis any family of continuous mappingsfWX!X; 2, where is a finite nonempty set (see[9]).

Let^Zdenote the set of all infinite sequencesfigi2Z of symbols belonging to. A typical element of^Z can be denoted asD f:::; 1; 0; 1; :::gand we use the shorted notation

F0Did;

FnDf_{n 1}of_{n 2}o:::of₀; F _nDf¹

nof¹

.n 1/o:::of ¹

1:

Please note that iff is a homeomorphism map for all2, then for everyn2Z and2^Z,F_nis a homeomorphism map onX.

A sequencefxngⁿ2Z inX is called an orbit of theIFSF if there exist 2^Zsuch thatxnC1Df_n.xn/, for eachn2.

TheIFSF D fXIfj2gisuniformly expandingif there exists ˇD inf

2inf

x¤y

d.f.x/; f.y//

d.x; y/

and this number called also theexpanding ratio, is greater than one [9].

We say thatF is expansive if there exist ae > 0such that for every arbitrary2^Z, d.Fi.x/;Fi.y// < e, for alli2Z, implies thatxDy.

Remark1. LetF be an uniformly expandingIFSandˇ > 1is it’s expanding ratio number. Suppose that for every2^Z,Fnand´2X,fFn.´/gⁿ2Z, is an infinite set.

Consider the number 1 < < ˇ, since is a finite set there exists at most a fi- nite set of pointsx; y2X such that ^d.f_d.x;y/^.x/;f.y// , for some2. Now, Let D f:::; 1; 0; 1; :::g be an element of ^Z and consider distinct points x; y 2 X. Suppose that d.Fi.x/;Fi.y// < 1 for all i 2Z. Since every orbit is an infinite set and f_i is injective, for all i 2 Z, then there exists k > 0 such that

d.f_k

Ci.F_k

Ci 1.x//;f_k

Ci.F_k

Ci 1.y///

d.F_k

Ci 1.x/;F_k

Ci 1.y// > , for alli0. So,ⁱd.Fk.x//;Fk.y//

< d.FkCi.x/;FkCi.y// < 1, for alli > 0, and consequentlyxDy. Then uniformly expanding implies the expansivity.

Also following example shows that expansivity does not imply uniformly expand- ing.

Example1. Let˙ denote the set of all bi-infinite sequence

xD. ; x 2; x 1; x0; x1; x2; / where xnD0 or1. The set ˙ becomes a com- pact metric space if we define the distance between two points x; y by .x; y/D P₁

iD 1jxi yij

2^jij . This is well known that he shift map g W˙ !˙ defined by . .x//i DxiC1is a homeomorphism and so by compactness of˙ is not uniformly expanding. Ifx¤y are two point in˙ then, for somek2Z,xk ¤yk and hence .g^k.x/; g^k.y//1, this implies thatgW˙ !˙is an expansive homeomorphism.

Letf W˙ !˙be a map defined by.f .x//iDxiC2, using the above arguments, we can prove that ifx¤yare two point in˙then, for somek2Z,.f^k.x/; f^k.y//

1

2. Now consider the IFSF D f˙Ifi ji 2 f1; 2gg, wheref1Dg andf2Df. If x ¤y are two point in ˙ then, for every 2^Z; there exists k 2Z such that .Fk.x/;Fk.y// ¹₂. So the IFS F D f˙Ifi ji 2 f1; 2gg is an expansive IFS contains homeomorphism functions and is not uniformly expanding.

Given ı > 0, a sequencefxigi2Z inX is called aı pseudo orbit ofF if there exist2^Zsuch that for everyi 2, we haved.xiC1; f_i.xi// < ı.

One says that theIFSF has theshadowing property if, given > 0, there existsı > 0 such that for anyı pseudo orbitfxigⁱ2Z there exist an orbitfyigⁱ2Z, satisfying the inequality d.xi; yi/ for alli2Z. In this case one says that thefyigi2Z or the pointy0, shadows theı pseudo orbitfxigⁱ2Z[9].

Please note that ifis a set with one member then theIFSF is an ordinary discrete dynamical system. In this case the shadowing property forF is ordinary shadowing

property for a discrete dynamical system.

The point y0 in definition of shadowing property related to the sequence which makes the orbitfyigi2Zis unique.

Remark2. Suppose that theIFSF satisfied shadowing property and is expansive with expansive constante > 0. Consider0 < <^e₃ and chooseı > 0corresponding to in the definition of shadowing property. Letfxigⁱ2Z be a ı pseudo orbit for IFS, there exist a sequencesD f:::; 1; 0; 1; :::gandfyigi2Z such thatyiC1D fi.yi/andd.xi; yi/ < , for all i2Z. Iff´igⁱ2Z be another sequence such that

´iC1Df_i.´i/andd.xi; ´i/ < , for alli2Z, then for alli2Z:

d.Fi.y0/;Fi.´0//Dd.yi; ´i/d.yi; xi/Cd.xi; ´i/ < 2 < e:

Thus by expansivity of theIFSF we have thaty₀D´₀.

We say thatF has theLipschitz shadowing propertyif there areL > 0and0> 0 such that for any0 < < 0and any pseudo orbitfxigi2ZofF there existy2X and2^Zsuch thatd.F_i.y/; xi/ < L, for alli2Z.

We say that F has the limit shadowing propertyif: for any sequence fxigi2Z of points inX, if limi!˙1d.f_i.xi/; xiC1/D0, for some

D f:::; 1; 0; 1; :::g 2^Zthen there is an orbitfyigi2Zsuch that limi!˙1d.xi; yi/D0

LetX^Z be the set of all sequencesfxigi2Zof points inX and leted be the metric on X^Z defined by

ed .fxigi2ZI fyigi2Z/Dsupi2Z

d.xiIyi/ 2^jij ;

forfxigⁱ2Z;fyigⁱ2Z2X^Z. Letp.F; ı/be the set of allı-pseudo-orbits.ı > 0/of F with the subspace topology ofX^Z[14].

We say thatF has thecontinuous shadowing propertyif for every > 0, there are a ı > 0and a continuous maprWp.F; ı/!X such that

d.Fi.r.x//; xi/ < , whereD f:::; 1; 0; 1; :::g,xD fxigⁱ2Z and d.f_i.xi/; xiC1/ < ı, for alli2Z.

3. RESULTS

By Theorem2, Sakai showed that any positively expansive open map has the shad- owing property. In this section we introduce openIFSand show that for an expansive IFS, the openness; shadowing property and Lipschitz shadowing property are equi- valent.

F D fXIfj2gis said to be an openIFSiffis an open map, for all2. Definition 1([10]). Letf WX !X be a continuous map on a compact metric space. We say that f expands small distances if there exist constants ı0> 0 and

˛ > 1such that0 < d.x; y/ < ı0(x; y2X) implies d.f .x/; f .y// > ˛d.x; y/.

We say thatF expands small distance, if there are constantsı0> 0and˛ > 1such thatd.f.x/; f.y// > ˛d.x; y/whenever0 < d.x; y/ < ı0.2/

Remark3. Suppose thatF D fXIfj2gis an expansive IFS, thenfis an expansive function and by Lemma 1. of [10] expand small distance. Let in the proof of Lemma 1. of [10]

VnD f.x; y/2XX Wd.F_i.x/;F_i.y//c; f or al l 2^Zand al l jij< ng: So,F expand small distance.

To prove Theorem3, we need the following lemma.

Lemma 1. Suppose thatF expands small distance with related constants

˛ > 1andı0> 0, then the following are equivalent:

i /F is an open IFS.

i i /There exists0 < ı1< ^ı₂⁰ such that ifd.f.x/; y/ < ˛ı1then

B_ı1.x/\f ¹.y/¤¿, for all2, whereB_ı1.x/is the neighborhood ofx with radiusı1.

Proof. Since everyfexpands small distance then by Lemma 2. of [10], for every 2the following are equivalent:

i / fis an open map.

i i /there exists0 < ı< ^ı₂⁰ such that ifd.f.x/; y/ < ˛ıthen B_ı.x/\f¹.y/¤¿.

Because of the proof of Lemma1in [5] for every2there exist infinitely0 < ı <

ı such thatd.f.x/; y/ < ˛ı impliesBı.x/\f¹.y/¤¿. So this sufficient to take

ı1DminfıW2g.

Corollary 1. Let the IFSF D fXIfj2gexpands small distance with related constants ˛ > 1 andı0> 0, and satisfied condition i i in Lemma 1. Then for all 0 < ı1,2andx; y2X,d.f.x/; y/ < implies thatB

˛.x/\f ¹.y/¤¿. Proof. Letd.f.x/; y/ < , since ı1 by Lemma 1. we have that B_ı1.x/\ f ¹.y/¤¿. Take´2B_ı1.x/\f¹.y/. This is clear thatyDf.´/andd.x; ´/ <

ı1:SinceF expands small distances,

˛d.x; ´/ < d.f.x/; f.´//Dd.f.x/; y/ < : Thusd.x; ´/ < _˛and henceB

˛.x/\f ¹.y/¤¿.

Theorem 3. Suppose thatF expands small distance with related constants

˛ > 1andı0> 0, the following conditions are equivalent:

i /F is an open IFS.

i i /F has the shadowing property.

i i i /F has the Lipschitz shadowing property.

Proof. .i i i)i i /By definitions of the shadowing and Lipschitz shadowing prop- erties this is clear that the Lipschitz shadowing property implies the shadowing prop- erty.

.i H)i i i /LetLD˛ 1^2˛ D2˙_k¹_D0˛ ^k> 1and fix any0 < <^ı_L¹, whereı1and˛ be as in Lemma1. Suppose thatfxigi2Z is an pseudo orbit forF; there is D f:::; 1; 0; 1; :::g 2^Zsuch thatd.f_i.xi/; xiC1/ < for alli2Z.

Pick anyi1and put˛jD˙_k^{j 1}_D0˛ ^kforj1. Sinced.f_i.xi/; xiC1/ < then, by Lemma1and Corollary1, there existsy_{i 1}^{.i /} 2B

˛.xi 1/such thatf_{i 1}.y_{i 1}^{.i /} /Dxi. Thus

d.f_{i 2}.xi 2/; y_{i 1}^{.i /} /d.f_{i 2}.xi 2/; xi 1/Cd.xi 1; y_{i 1}^{.i /} /

< C

˛ D.1C1

˛/ < L:

Hence there existsy_{i 2}^{.i /} 2B_˛2

˛.xi 2/such thatf_{i 2}.y_{i 2}^{.i /} /Dy_{i 1}^{.i /} and so d.f_{i 2}.xi 3/; y_{i 2}^{.i /} /d.f_{i 3}.xi 3/; xi 2/Cd.xi 2; y_{i 2}^{.i /} /

< C˛2

˛ < ˛3 < L:

Because of Lemma1and Corollary1there existsy_{i 3}^{.i /} 2B_˛3

˛.xi 3/such that f_{i 3}.y_{i 3}^{.i /} /Dy_{i 2}^{.i /} . Thusd.f_{i 3}.xi 4/; y_{i 3}^{.i /} / < ˛4 < L.

Repeating the process, we can find:

y₀^{.i /}2B_˛i

˛.x0/such thatf₀.y₀^{.i /}/Dy₁^{.i /}, y^{.i /}₁2B_˛iC1

˛.x0/such thatf ₁.y^{.i /}₁/Dy₀^{.i /}, :::

y^{.i /}_i 2B_˛2i

˛.x0/such thatf _i.y^{.i /}_i/Dy^{.i /}_iC1.

SinceXis compact, if we letykDlimi!1y_k^{.i /}, thenf_k.yk/DykC1andd.yk; xk/

< L, for allk2Z. ThereforeF has the Lipschitz shadowing property.

.i i)i /. SinceF has the shadowing property, there exist0 < ı <^ı₂⁰ such that every ı˛ pseudo orbit ofF isı0 shadowed by some point. Now, fix2. Consider x; y2X such thatd.f.x/; y/ < ı˛and define aı˛ pseudo orbit ofF byx0Dx andxi Df^{i 1}.y/ .i 2Z/. Then there exists´2X and D f:::; 1; 0; 1; :::g 2 ^Zsuch thatd.F_i.´/; xi/ < ı0, for all

i 2Z. Less of generality; by proof of Theorem 2.2. in [9] and this fact thatxiC1D f.xi/ .i2Z/, we can assume that thatiDfor alli0. Then˛^{i 1}d.f.´/; y/

d.fⁱ.´/; f^{i 1}.y//ı0 for all i0, sof.´/Dy. This implies that´Df ¹.y/

andd.x; ´/ < ı0, then

d.x; ´/ < ^d.f.x/;f_˛ ^.´// D ^.f.x/;y/_˛ < ^ı˛_˛. Hence B_ı.x/\f ¹.y/¤¿. So, by

Lemma1F is an open IFS.

The next theorem is one of the main results of this paper and demonstrates that for an expansiveIFS, the limit shadowing property and the shadowing property are equivalent.

Theorem 4. Let X be a compact metric space and F D fXIfj2gbe an expansive IFS onZ. The following conditions are equivalent:

i /F has the shadowing property,

i i / there is a compatible metric D for X such that F has the limit shadowing property with respect toD.

Proof. By definitions the assertion.i i)i /is clear.

To prove.i)i i /, at first we have the following lemmas.

Lemma 2. There is a compatible metricDonX andK1such that D.f.x/; f.y//KD.x; y/;

D.f ¹.x/; f ¹.y//KD.x; y/

for anyx; y2X and2.

Proof. SinceF is expansive thenf is expansive, for every2. So, by [10]

(page 3) for every2there existsK> 1such that D.f.x/; f.y//KD.x; y/;

D.f ¹.x/; f¹.y//KD.x; y/

for anyx; y2X. TakeKDmaxfKW2g, the proof is complete.

To prove.i )i i /we need to define the local stable set and the local unstable set for anIFS.

Let > 0,2^Zandxbe an arbitrary point ofX then W^s₀.x; /D fyId.Fn.x/;Fn.y//; 8n0g, W^u₀.x; /D fyId.F n.x/;F n.y//; 8n > 0g

is said to be the local stable set and the local unstable set ofxrespect to2^Z. Lemma 3. There exist constants0> 0and < 1such that

D.Fi.x/;Fi.y//ⁱD.x; y/ if y2W^s₀.x; /;

D.F _i.x/;F _i.y//ⁱD.x; y/ if y2W^u

0.x; /

Proof. Since for every2,fis an expansive map, To proof the lemma this is sufficient to in Lemma 1 of [13] we assume that

WnD f.x; y/2XXWd.F_i.x/;F_i.y//c; f or al l ji j< ng:

The rest of proof is similar to [13].

.i)i i /LetDbe the compatible metric forX by the above lemmas.

First, notice that iff´igⁱ2Zis an pseudo orbit ofF,. _2L⁰/i.e.

D.f_i.´i/; ´iC1/ < , for someD f:::; 1; 0; 1; :::g 2^Z. Then by Theorem3, there existsy2X such thatD.Fi.y/; ´i/ < Lfor alli 2Z.

Now, suppose thatfxigⁱ2Zis a sequence of points inX such that

limi!˙1D.f_i.xi/; xiC1/D0, for some D f:::; 1; 0; 1; :::g 2^Z. For any ı > 0 .ı <_2L⁰/;there existsIı> 0such thatjij> Iıimplies thatD.f_i.xi/; xiC1/ <

ı. Note that

f:::; f ¹

Iı 2.f ¹

Iı 1.x_Iı//; f¹

Iı 1.x_Iı/; x_Iı; x_IıC1; :::g is aı pseudo orbit ofF, and by Theorem3there existsy_ı2X such that

D.F_i.y_ı/; xi/ < Lıfor alliI_ı. By the same way, there exists´_ı2X;Such that D.F i.´ı/; x i/ < Lıfor alliIı. Thus:

D.F_i.y/;F_i.y_ı//D.F_i.y/; xi/CD.xi;F_i.y_ı// < 0

for alli I_ı. This implies that F_Iı.y_ı/2W^s

0.F_Iı.y//. So that, by Lemma 3, D.Fi.y_ı/;Fi.y//^{i Iı}D.F_Iı.y_ı/;F_Iı.y// for all i I_ı. Mimicking the procedure, we have D.F _i.y_ı/;F _i.y//^{i Iı}D.F _Iı.y_ı/;F _Iı.y//for all iIı. TakeJı> Iı such that0^{i Iı} < ıifiJı. Since

D.F_i.y/; xi/D.F_i.y/;F_i.y_ı//CD.F_i.y_ı/; xi/ and

D.F i.y/; x i/D.F i.y/;F i.y_ı//CD.F i.y_ı/; x i/:

It is easy to see that

maxfD.F_i.y/; xi/; D.F _i.y/; x i/g< .LC1/ı. Thus

limi!˙1D.Fi.y/; xi/D0.

Fix ı > 0 and 2. Suppose that fxigⁱ2ZC is a ı pseudo orbit for F and considerfyigi2Z as the following:

yiD

xi if i0;

fⁱ.x0/ if i < 0:

So,fyigⁱ2Z is aı pseudo orbit forF. Then shadowing properties onZimplies the shadowing properties onZ_C.

By Theorems3,4and Theorem 3.2. of [6] we have the following corollary.

Corollary 2. Let X be a compact metric space. If F D fXIfj2g is an expansive IF S with the limit (Lipschitz) shadowing property ( onZ_C), then so is F ¹D fXIgj2gwherefWX!X is homeomorphism andgDf ¹for all 2.

By Theorems3,4and Theorem 3.5. of [6] we have the following corollary.

Corollary 3. Letbe a finite set,F D fXIfj2gis anIF S and letk > 0 be an integer. SetF^kD fgj2˘g D ff_ko:::of₁j1; :::; _k 2g.

IfF has the limit (Lipschitz) shadowing property ( onZ_C), then so doesF^k. We say thatF is strongly expansive if there exist a metricd forX and a constant e0such that for every two arbitrary; 2^Z,

d.F_i.x/;F_i.y// < e, for alli2Z, implies thatxDy.

To prove Theorem5, we need the following lemma.

Lemma 4. Suppose thatF is an expansiveIFSwith expansive constante,˛is an arbitrary positive number and; 2^Z. For allx; y2X there exists an integer N DN.e; ˛/ > 0such that ifd.Fi.x/;Fi.y//efor allji jN, thend.x; y/ <

˛.

Proof. We will give a proof by contradiction. Suppose that for each

n 1, there exist xn and yn with d.Fi.xn/;Fi.yn// e for all ji jn and d.xn; yn/˛. SinceX is a compact metric space, we may assume thatxn!xand yn!y and hence thatF_i.xn/!F_i.x/F_i.yn//!F_i.y/for everyji j1.

Then d.Fi.x/;Fi.y//e for all i 2Zandd.x; y/ ˛, which is a contradic-

tion.

Theorem 5. Let X be a compact metric space and F D fXIfj2gbe an strongly expansive IFS onZ. The following conditions are equivalent:

i /F has the shadowing property,

i i /F has the continuous shadowing property.

Proof. By definitions, this is clear that continuous shadowing property implies the shadowing property.

.i )i i /. Lete > 0be an expansive constant ofF. For0 < < ^e₃, letıDı./ <

be as in the shadowing property of F. It is easy to see that for any ı pseudo orbitfxigⁱ2Z ofF by Remark2 there exists D f:::; 1; 0; 1; :::g 2^Z and a uniquey2Xsatisfyingd.F_i.y/; xi/ < for alli2Z. So we definerWP .F; ı/! X by r.fxigi2Z/Dy. To show that the mapr is continuous, choose an arbitrary constant˛ > 0, by Lemma4 there exists an integer N DN.e; ˛/ > 0such that if d.Fi.y/;Fi.y⁰//efor; 2^Zand alljijN, thend.y; y⁰/ < ˛. Pickˇ > 0 such that2^Nˇ < ^e₃. Letfxigⁱ2Z,fx_i⁰gⁱ2Z2P .F; ı/be given twoı pseudo orbit of F with related sequences; 2^Zand letr.fxigi2Z/Dyandr.fx⁰_igi2Z/Dy⁰. ifd.fxigⁱ2Z;fx_i⁰gⁱ2Z/ < ˇ, then we see that d.xi; x_i⁰/ < ^e₂ for allji j< N, so that d.Fi.y/;Fi.y⁰//d.Fi.y/; xi/Cd.xi; x⁰_i/Cd.Fi.y⁰/; x_i⁰/ < efor alljijN. Thusd.y; y⁰/ < ˛by the choice ofN, and the conclusion is obtained.

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Author’s address

Mehdi Fatehi Nia

Yazd University, Department of Mathematics, 89195-741 Yazd, Iran E-mail address:fatehiniam@yazd.ac.ir