Various shadowing properties and their equivalent for expansive iterated function systems

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by  Mehdi Fatehi Nia

VARIOUS SHADOWING PROPERTIES AND THEIR EQUIVALENT FOR
EXPANSIVE ITERATED FUNCTION SYSTEMS
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MEHDI FATEHI NIA
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2 DEPARTMENTOFMATHEMATICS,YAZDUNIVERSITY,89195-741YAZD,IRAN

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E-MAIL:[email protected]
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J

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S Abstract. In this paper we introduce expansive iterated function systems, ( IFS) on
D
a compact metric space then various shadowing properties and their equivalence are
.
h
t consideredforexpansiveIFS.
a
m
keywords: ExpansiveIFS,pseudoorbit,shadowing,continuousshadowing,limitshadow-
[

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ing, Lipschitz shadowing.
v
7
3 subjclass[2010] 37C50,37C15
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0 1. Introduction
.
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The notion of shadowing plays an important role in dynamical systems, specially; in
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1
: stabilitytheory[1,9,11]. Variousshadowingpropertiesforexpansivemapsandtheirequiv-
v
i
X alence havebeen studiedbyLee andSakai[10,13]. Moreprecisely,they provethe following
r
a
theorems:
Theorem1.1. [10]Letf beanexpansivehomeomorphism onacompactmetricspace(X;d).
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 metric D for X such that f has the Lipschitz shadowing property
with respect to D,
(d) f has the limit shadowing property,
1
2 MEHDIFATEHINIA
(e) there is a compatible metric D for X such that f has the strong shadowing property with
respect to D.
Theorem 1.2. [13] Let f be a positively expansive map on a compact metrizable space X.
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 that f has the Lipschitz shadowing property,
(d) there is a metric such that f has the s-limit shadowing property,
(e) there is a metric such that f has the strong shadowing property.
Intheotherhand,iteratedfunctionsystems(IFS),areusedfortheconstructionofdeter-
ministic fractals and have found numerous applications, in particular to image compression
and image processing [2]. Important notions in dynamics like attractors, minimality, tran-
sitivity, and shadowing can be extended to IFS (see [3, 4, 7, 8]). The authors defined the
shadowingpropertyforaparameterizediteratedfunctionsystemandprovethatifaparam-
eterizedIFSisuniformlyexpanding(orcontracting),thenithastheshadowingproperty[7].
In this paper we present an approach to shadowing property for iterated function systems.
At first, we introduce expansive iterated function systems on a compactmetric space. Then
continuous shadowing, limit shadowing and Lipschitz shadowing properties are defined for
an IFS, F = {X;f |λ ∈ Λ} where Λ is a nonempty finite set and f : X → X is home-
λ λ
omorphism, for all λ ∈ Λ. Theorems 3.4 and 3.5 are the main result of the present work.
Actually in these theorems we prove that the limit shadowing property, the Lipschitz shad-
owingpropertyareallequivalenttotheshadowingpropertyforexpansiveIFSonacompact
metric space. The method is essentially the same as that used in [10, 12, 13]. Finally, we
introducethestrongexpansiveIFSandshowthatforastrongexpansiveIFSthecontinuous
shadowing property and the shadowing property are equivalent.
VARIOUS SHADOWING PROPERTIES AND THEIR EQUIVALENT FOR... 3
2. preliminaries
Inthis section,wegivesomedefinitionsandnotationsaswellassomepreliminaryresults
that are needed in the sequel. Let (X,d) be a complete metric space. Let us recall that an
Iterated Function System(IFS) F = {X;f |λ ∈ Λ} is any family of continuous mappings
λ
f :X →X, λ∈Λ, where Λ is a finite nonempty set (see[7]).
λ
Let ΛZ denote the set of all infinite sequences {λi}i∈Z of symbols belonging to Λ. A typical
element of ΛZ can be denoted as σ ={...,λ ,λ ,λ ,...} and we use the shorted notation
−1 0 1
F =id,
σ0
F =f of o...of ,
σn λn−1 λn−2 λ0
F =f−1 of−1 o...of−1 .
σ−n λ−n λ−(n−1) λ−1
Please note that if f is a homeomorphism map for all λ ∈ Λ, then for every n ∈ Z and
λ
σ ∈ΛZ, F is a homeomorphism map on X.
σn
A sequence {xn}n∈Z in X is called an orbit of the IFS F if there exist σ ∈ ΛZ such that
x =f (x ), for each λ ∈σ.
n+1 λn n n
The IFS F ={X;f |λ∈Λ} is uniformly expanding if there exists
λ
d(f (x),f (y))
λ λ
β =sup sup
λ∈Λ x6=y d(x,y)
and this number called also the expanding ratio, is greater than one [7].
We say that F is expansive if there exist a e > 0 such that for every arbitrary σ ∈ ΛZ,
d(F (x),F (y))<e, for all i∈Z, implies that x=y.
σi σi
Remark 2.1. Let F be an uniformly expanding IFS and β > 1 is it’s expanding ratio
number. Suppose that σ ∈ ΛZ and d(F (x),F (y)) < 1 for all i ∈ Z. So, βid(x,y) <
σi σi
4 MEHDIFATEHINIA
d(F (x),F (y)) < 1, for all i > 0, and consequently x = y. Then uniformly expanding
σi σi
implies the expansivity.
Givenδ >0,asequence{xi}i∈Z inX iscalledaδ−pseudoorbitofF ifthereexistσ ∈ΛZ
such that for every λ ∈σ, we have d(x ,f (x ))<δ.
i i+1 λi i
OnesaysthattheIFS F hastheshadowing property if,givenǫ>0,thereexistsδ >0such
that for any δ−pseudo orbit {xi}i∈Z there exist an orbit {yi}i∈Z, satisfying the inequality
d(xi,yi)≤ǫ for all i∈Z. Inthis case one saysthat the {yi}i∈Z or the point y0, ǫ−shadows
the δ−pseudo orbit {xi}i∈Z[7].
Please note that if Λ is a set with one member then the IFS F is an ordinary discrete
dynamicalsystem. InthiscasetheshadowingpropertyforF isordinaryshadowingproperty
for a discrete dynamical system.
Remark 2.2. If the IFS F is expansivewith expansive constante>0 and0<ǫ< e then
3
the point y in the above definition is unique.
0
We say that F has the Lipschitz shadowing property if there are L > 0 and ǫ > 0 such
0
that for any 0<ǫ<ǫ0 and any ǫ−pseudo orbit {xi}i∈Z of F there exist y ∈X and σ ∈ΛZ
such that d(F (y),x )<Lǫ, for all i∈Z.
σi i
We say that F has the limit shadowing property if: for any sequence {xi}i∈Z of points in
X, if lim d(f (x ),x ) = 0, for some σ = {...,λ ,λ ,λ ,...} ∈ ΛZ then there is an
i→±∞ λi i i+1 −1 0 1
orbit {yi}i∈Z such that limi→±∞d(xi,yi)=0
Let XZ be the set of all sequences {xi}i∈Z of points in X and let d be the metric on XZ
e
defined by
d(x ;y )
d({xi}i∈Z;{yi}i∈Z)=supi∈Z 2i|i| i ,
e
for {xi}i∈Z,{yi}i∈Z ∈ XZ. Let p(F,δ) be the set of all δ-pseudo-orbits (δ > 0) of F with
the subspace topology of XZ [10].
VARIOUS SHADOWING PROPERTIES AND THEIR EQUIVALENT FOR... 5
We say that F has the continuous shadowing property if for every ǫ > 0, there are a δ > 0
and a continuous map r :p(F,δ)→X such that d(F (r(x)),x )<ǫ, where
σi i
σ ={...,λ−1,λ0,λ1,...}, x={xi}i∈Z and d(fλi(xi),xi+1)<δ, for all i∈Z.
3. Results
ByTheorem1.2,Sakaishowedthatanypositivelyexpansiveopenmaphastheshadowing
property. In this section we introduce open IFS and show that for an expansive IFS, the
openness; shadowing property and Lipschitz shadowing property are equivalent.
F ={X;f |λ∈Λ} is said to be an open IFS if f is an open map, for all λ∈Λ.
λ λ
Definition 3.1. [13] Let f :X →X be a continuous map on a compact metric space. We
say that f expands small distances if there exist constants δ > 0 and α > 1 such that
0
0<d(x,y)<δ (x,y ∈X) implies d(f(x),f(y))>αd(x,y).
0
We say that F expands smalldistance, if there are constants δ >0 and α>1 suchthat
0
d(f (x),f (y))>αd(x,y) whenever 0<d(x,y)<δ (λ∈Λ)
λ λ 0
Remark3.2. SupposethatF ={X;f |λ∈Λ}isanexpansiveIFS,thenf isanexpansive
λ λ
function and by Lemma 1. of [13] expand small distance. Let in the proof of Lemma 1. of
[13]
Z
V ={(x,y)∈X ×X :d(F (x),F (y))≤c,for all σ ∈Λ and all |i|<n}.
n σi σi
So, F expand small distance.
To prove Theorem 3.4, we need the following lemma.
Lemma 3.3. Suppose that F expands small distance with related constants α > 1 and
δ >0, then the following are equivalent:
0
i) F is an open IFS.
6 MEHDIFATEHINIA
ii) There exists 0<δ1 < δ20 such that if d(fλ(x),y)<αδ1 then Bδ1(x)∩fλ−1(y)6=∅, for all
λ∈Λ, where B (x) is the neighborhood of x with radius δ .
δ1 1
Proof. Since everyf expandssmalldistance thenby Lemma 2. of[13],for everyλ∈Λ the
λ
following are equivalent:
i) f is an open map.
λ
ii) there exists 0<δλ < δ20 such that if d(fλ(x),y)<αδλ then Bδλ(x)∩fλ−1(y)6=∅.
Becauseofthe proofofLemma1in[5]foreveryλ∈Λthere existinfinitely 0<δ <δ such
λ
that d(f (x),y)<αδ implies B (x)∩f−1(y)6=∅. So this sufficient to take
λ δλ λ
δ =min{δ :λ∈Λ}. (cid:3)
1 λ
Theorem 3.4. Under the above assumption, the following conditions are equivalent:
i) F is an open IFS.
ii) F has the shadowing property.
iii) F has the Lipschitz shadowing property.
Proof. (iii⇒ii)By definitions of the shadowingand Lipschitz shadowing properties this is
clear that the Lipschitz shadowing property implies the shadowing property.
(i =⇒ iii) Let L = 2α = 2Σ∞ α−k > 1 and fix any 0 < ǫ < δ1, where δ and α be as in
α−1 k=0 L 1
Lemma 3.3. Suppose that {xi}i∈Z is an ǫ−pseudo orbit for F; there is
σ ={...,λ ,λ ,λ ,...}∈ΛZ such that d(f (x ),x )<ǫ for all i∈Z.
−1 0 1 λi i i+1
Pickanyi≥1 andput α =Σj−1α−k forj ≥1. Sinced(f (x ),x )<ǫ then,by Lemma
j k=0 λi i i+1
3.3, there exists yi(−i)1 ∈Bαǫ(xi−1) such that fλi−1(yi(−i)1)=xi. Thus
d(f (x ),y(i) )≤d(f (x ),x )+d(x ,y(i) )<ǫ+ ǫ =ǫ(1+ 1)<ǫL.
λi−2 i−2 i−1 λi−2 i−2 i−1 i−1 i−1 α α
Hence there exists yi(−i)2 ∈Bα2αǫ(xi−2) such that fλi−2(yi(−i)2)=yi(−i)1 and so
d(f (x ),y(i) )≤d(f (x ),x )+d(x ,y(i) )<ǫ+α ǫ <α ǫ<ǫL.
λi−2 i−3 i−2 λi−3 i−3 i−2 i−2 i−2 2α 3
Because of Lemma 3.3 there exists yi(−i)3 ∈ Bα3αǫ(xi−3) such that fλi−3(yi(−i)3) = yi(−i)2. Thus
d(f (x ),y(i) )<α ǫ<ǫL.
λi−3 i−4 i−3 4
VARIOUS SHADOWING PROPERTIES AND THEIR EQUIVALENT FOR... 7
Repeating the process, we can find:
y0(i) ∈Bαiαǫ(x0) such that fλ0(y0(i))=y1(i),
y−(i)1 ∈Bαi+1αǫ(x0) such that fλ−1(y−(i)1)=y0(i),
.
.
.
y−(i)i ∈Bα2iαǫ(x0) such that fλ−i(y−(i)i)=y−(ii)+1.
Since X is compact, if we let y =lim y(i), then f (y )=y and d(y ,x )<ǫL, for
k i→∞ k λk k k+1 k k
all k ∈Z. Therefore F has the Lipschitz shadowing property.
(ii ⇒ i). Since F has the shadowing property, there exist 0 < δ < δ0 such that every
2
δα−pseudo orbit of F is δ −shadowed by some point. Now, fix ν ∈ Λ. Consider x,y ∈
0
X such that d(f (x),y) < δα and define a δα−pseudo orbit of F by x = x and x =
ν 0 i
fi−1(y) (i ∈ Z). Then there exists z ∈ X and σ = {...,λ ,λ ,λ ,...} ∈ ΛZ such that
ν −1 0 1
d(F (z),x ) < δ , for all i ∈ Z. Less of generality; by proof of Theorem 2.2. in [7]
σi i 0
and this fact that x = f (x ) (i ∈ Z), we can assume that that λ = ν for all i ≥ 0.
i+1 ν i i
Then αi−1d(f (z),y) ≤ d(fi(z),fi−1(y)) ≤ δ for all i ≥ 0, so f (z) = y. This implies
ν ν ν 0 ν
that z = f−1(y) and d(x,z) < δ , then d(x,z) < d(fν(x),fν(z)) = (fν(x),y) < δα. Hence
ν 0 α α α
B (x)∩f−1(y)6=∅. So, by Lemma 3.3 F is an open IFS. (cid:3)
δ ν
The next theorem is one of the main results of this paper and demonstrates that for an
expansive IFS, the limit shadowing property and the shadowing property are equivalent.
Theorem 3.5. Let X be a compact metric space and F = {X;f |λ ∈ Λ} be an expansive
λ
IFS on Z. The following conditions are equivalent:
i) F has the shadowing property,
ii) there is a compatible metric D for X such that F has the limit shadowing property with
respect to D.
Proof. By definitions the assertion (ii⇒i) is clear.
8 MEHDIFATEHINIA
To prove (i⇒ii), at first we have the following lemmas.
Lemma 3.6. There is a compatible metric D on X and K ≥1 such that
 D(f (x),f (y))≤KD(x,y),
λ λ

D(f−1(x),f−1(y))≤KD(x,y)
 λ λ
for any x,y ∈X and λ∈Λ.
Proof. Since F is expansive then f is expansive, for every λ ∈ Λ. So, by [13] (page 3) for
λ
every λ∈Λ there exists K >1 such that
λ
 D(f (x),f (y))≤K D(x,y),
λ λ λ

D(f−1(x),f−1(y))≤K D(x,y)
 λ λ λ
for any x,y ∈X. Take K =max{K :λ∈Λ}, the proof is complete. (cid:3)
λ
To prove (i⇒ii) we need to define the local stable set and the local unstable set for an
IFS.
Let ǫ>0, σ ∈ΛZ and x be an arbitrary point of X then
Ws(x,σ)={y;d(F (x),F (y))≤ǫ, ∀n≥0},
ǫ0 σn σn
Wu(x,σ)={y;d(F (x),F (y))≤ǫ, ∀n>0}
ǫ0 σ−n σ−n
is said to be the local stable set and the local unstable set of x respect to σ ∈ΛZ.
Lemma 3.7. There exist constants ǫ >0 and η <1 such that
0
 D(F (x),F (y))≤ηiD(x,y) if y ∈Ws(x,σ),
 σi σi ǫ0
D(F (x),F (y))≤ηiD(x,y) if y ∈Wu(x,σ)
 σ−i σ−i ǫ0
VARIOUS SHADOWING PROPERTIES AND THEIR EQUIVALENT FOR... 9
Proof. Since for every λ∈Λ, f is an expansive map, To proof the lemma this is sufficient
λ
to in Lemma 1 of [12] we assume that
W ={(x,y)∈X×X :d(F (x),F (y))≤c,for all |i|<n}.
n σi σi
The rest of proof is similar to [12]. (cid:3)
(i⇒ii)Let D be the compatible metric for X by the above lemmas. Let{xi}i∈Z be any
ǫ−pseudo orbit of F, (ǫ≤ 2ǫL0) i.e. D(fλi(xi),xi+1)<ǫ, for some
σ = {...,λ ,λ ,λ ,...} ∈ ΛZ. Then by Theorem 3.4, there exists y ∈ X such that
−1 0 1
D(F (y),x ) < Lǫ for all i ∈ Z. Suppose further that lim D(f (x ),x ) = 0. For
σi i i→±∞ λi i i+1
any δ >0 (δ < 2ǫL0), there exists Iδ >0 suchthat |i|>Iδ implies that D(fλi(xi),xi+1)<δ.
Note that
{...,f−1 (f−1 (x )),f−1 (x ),x ,x ,...}
λIδ−2 λIδ−1 Iδ λIδ−1 Iδ Iδ Iδ+1
isaδ−pseudoorbitofF,andbyTheorem3.4thereexistsy ∈X suchthatD(F (y ),x )<
δ σi δ i
Lδ for all i ≥ I . By the same way, there exists z ∈ X, Such that D(F (z ),x ) < Lδ
δ δ σ−i δ −i
for all i≥I . Thus:
δ
D(F (y),F (y ))≤D(F (y),x )+D(x ,F (y ))<ǫ
σi σi δ σi i i σi δ 0
for all i ≥ I . This implies that F (y ) ∈ Ws(F (y)). So that, by Lemma 3.7,
δ σIδ δ ǫ0 σIδ
D(F (y ),F (y)) ≤ ηi−IδD(F (y ),F (y)) for all i ≥ I . Mimicking the procedure,
σi δ σi σIδ δ σIδ δ
we have D(F (y ),F (y)) ≤ ηi−IδD(F (y ),F (y)) for all i ≥ I . Take J > I
σ−i δ σ−i σ−Iδ δ σ−Iδ δ δ δ
such that ǫ ηi−Iδ <δ if i≥J . Since
0 δ
D(F (y),x )≤D(F (y),F (y ))+D(F (y ),x )
σi i σi σi δ σi δ i
10 MEHDIFATEHINIA
and
D(F (y),x )≤D(F (y),F (y ))+D(F (y ),x ).
σ−i −i σ−i σ−i δ σ−i δ −i
It is easy to see that
max{D(F (y),x ),D(F (y),x )}<(L+1)δ. Thus lim D(F (y),x )=0. (cid:3)
σi i σ−i −i i→±∞ σi i
Fix δ > 0 and λ ∈ Λ. Suppose that {xi}i∈Z+ is a δ−pseudo orbit for F and consider
{yi}i∈Z as the following:
 x if i≥0,
i
yi =
fi(x ) if i<0.
 λ 0
So,{yi}i∈Zisaδ−pseudoorbitforF. ThenshadowingpropertiesonZimpliestheshadowing
properties on Z .
+
By Remark 3.2, Theorems 3.4, 3.5 and Theorem 2.2. of [7] we have the following corollary.
Corollary 3.8. If an IFS F ={X;f |λ∈Λ} is uniformly expanding and if each function
λ
f (λ∈Λ)ishomeomorphism, thentheIFShastheLipschitzshadowingandlimitshadowing
λ
properties on Z .
+
By Theorems 3.4, 3.5 and Theorem 3.2. of [6] we have the following corollary.
Corollary 3.9. Let X be a compact metric space. If F = {X;f |λ ∈ Λ} is an expansive
λ
IFS withthelimit(Lipschitz) shadowingproperty(onZ ),thensoisF−1 ={X;g |λ∈Λ}
+ λ
where f :X →X is homeomorphism and g =f−1 for all λ∈Λ.
λ λ λ
By Theorems 3.4, 3.5 and Theorem 3.5. of [6] we have the following corollary.
Corollary 3.10. Let Λ be a finite set, F = {X;f |λ ∈ Λ} is an IFS and let k > 0 be an
λ
integer. Set Fk ={g |µ∈Π}={f o...of |λ ,...,λ ∈Λ}.
µ λk λ1 1 k
If F has the limit (Lipschitz) shadowing property ( on Z ), then so does Fk.
+