# Thed-Shadowing Property and Average Shadowing Property for Iterated Function Systems

Authors: Jie Jiang; Lidong Wang; Yingcui Zhao
Journal: Complexity
 (2020)

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## Abstract

In this paper, we introduce the definitions ofd¯-shadowing property, d¯-shadowing property, topological ergodicity, and strong ergodicity of iterated function systems IFSf0,f1. Then, we show the following: 1 if IFSf0,f1 has the d¯-shadowing property (respectively, d¯-shadowing property), then ℱk has the d¯-shadowing property (respectively, d¯-shadowing property) for any k∈Z+; 2 if ℱk has the d¯-shadowing property (respectively, d¯-shadowing property) for some k∈Z+, then IFSf0,f1 has the d¯-shadowing property (respectively, d¯-shadowing property); 3 if IFSf0,f1 has the d¯-shadowing property or d¯-shadowing property, and f0 or f1 is surjective, then IFSf0,f1 is chain mixing; 4 let f0,f1 be open maps. For IFSf0,f1 with the d¯-shadowing property (respectively, d¯-shadowing property), if A⊂X is dense in X, and s is a minimal point of f0 or f1 for any s∈A, then IFSf0,f1 is strongly ergodic, and hence, ℱk is strongly ergodic; and 5 for IFSf0,f1 with the average shadowing property, if S⊂X is dense in X, and s is a quasi-weakly almost periodic point of f0 or f1 for any s∈S, then IFSf0,f1 is ergodic.


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## Body

## 1. Introduction

In this paper, letN=0,1,2,… and Z+=1,2,3,…. Suppose that X is a compact metric space and f:X⟶X a continuous map. The set J⊂N is a syndetic set if there is N0∈Z+ such that n,n+N0∩J≠∅ for each n∈N. For any x∈X,ε>0, let Bx,ε denote the ε-neighborhood of x. x∈X is a minimal point of f if for any neighborhood U of x, the set Nx,U=n∈N:fnx∈U is syndetic, and the set of all minimal points of f is denoted by APf. x∈X is called a quasi-weakly almost periodic point of f if for any neighborhood U of x, the set Nx,U=n∈N:fnx∈U has positive upper density.The shadowing property is a very important notion in dynamical systems. Many researchers have found some relationship among various shadowing properties, chain transitivity, transitivity, and ergodicity. Gu [1] proved that if X,f has the asymptotic average shadowing property, and f is surjective, then X,f is chain transitive. For more recent results about various shadowing properties, one can refer to [2–9] and references therein.Aδ-ergodic-pseudo-orbit of f is a sequence xii≥0 such that for any i∈N,(1)limn⟶∞1ni∈N:0≤i<n,dfxi,xi+1<δ=1,where ⋅ represents the cardinality.f is said to have the d¯-shadowing property if for any ε>0, there is δ>0 such that every δ-ergodic-pseudo-orbit xii≥0 is ε-shadowed by a true orbit fizi≥0 in a way such that(2)limsupn⟶∞1ni∈N:0≤i<n,dfiz,xi<ε>12,where f is said to have the d¯-shadowing property if for any ε>0, there is δ>0 such that every δ-ergodic-pseudo-orbit xii≥0 is ε-shadowed by a true orbit fizi≥0 in a way such that(3)lim infn⟶∞1ni∈N:0≤i<n,dfiz,xi<ε>0.LetX be a metric space and f0,f1 be continuous maps on X. The iterated function system IFSf0,f1 is the action of the semigroup generated by f0,f1 on X. In this paper, we introduce the definitions of the d¯-shadowing property and d¯-shadowing property for IFSf0,f1.Anorbit of IFSf0,f1 is a sequence fωixi≥0, where ω=ω0ω1ω2…∈Σ2=α=α0α1α2…:αi∈0,1, and for any i∈N,(4)fωix=fωi−1∘⋯∘fω0x,fω0x=x.A sequenceξii≥0 is called a δ-ergodic-pseudo-orbit for IFSf0,f1 if there is ω∈Σ2 such that(5)limn⟶∞1ni∈N:0≤i<n,dfωiξi,ξi+1<δ=1.IFS
f
0
,
f
1 is said to have the d¯-shadowing property if for any ε>0, there is δ>0 such that every δ-ergodic-pseudo-orbit xii≥0 is ε-shadowed by a true orbit fωizi≥0 in a way such that(6)lim supn⟶∞1ni∈N:0≤i<n,dfωiz,ξi<ε>12.IFS
f
0
,
f
1 is said to have the d¯-shadowing property if for any ε>0, there is δ>0 such that every δ-ergodic-pseudo-orbit ξii≥0 is ε-shadowed by a true orbit fωizi≥0 in a way such that(7)lim infn⟶∞1ni∈N:0≤i<n,dfωiz,ξi<ε>0.Denoteℱk=X:fωk−1∘⋯∘fω1∘fω0:ω0,ω1,…,ωk−1∈0,1, where k∈Z+.In this paper, the definitions of the shadowing property and average shadowing property forIFSf0,f1 are introduced by Bahabadi [10], and the definition of the asymptotic average shadowing property for IFSf0,f1 is introduced by Nia [11]. Let f0,f1 be open maps. For IFSf0,f1 with the d¯-shadowing property (respectively, d¯-shadowing property), if A⊂X is dense in X, and s is a minimal point of f0 or f1 for any s∈A, then IFSf0,f1 is strongly ergodic. Under similar conditions, Niu [2] researched that X,f has the average shadowing property and then the conclusion is true. Wang and Niu showed that, for X,f with the average shadowing property, if S⊂X is dense in X, and s is a quasi-weakly almost periodic point of f for any s∈S, then f is transitivity (see [12]). However, under similar conditions, we will prove that IFSf0,f1 is ergodic. Then, we will come up with a situation that IFSf0,f1 does not have the asymptotic average shadowing property.According to Bahabadi [10], a finite sequence ξ0=x,…,ξk=y is called a δ-chain of IFSf0,f1 if for any i=0,…,k−1, there is ωi∈0,1 such that dfωiξi,ξi+1<δ.Definition 1.
(see [10]). IFSf0,f1 is as follows:(1)
Chain transitive if for anyx,y∈X and any δ>0, there is a δ-chain of IFSf0,f1 from x to y(2)
Transitive if for any nonempty open setsU,V⊂X, there are ω∈Σ2 and n∈N such that fωnU∩V≠∅(3)
Mixing if for any nonempty open setsU,V⊂X, there are ω∈Σ2 and N∈N such that fωnU∩V≠∅ for any n≥N(4)
Chain mixing if for anyx,y∈X and any δ>0, there is N∈Z+ such that for any n≥N, there is a δ-chain of IFSf0,f1 from x to y consisting of exactly n elements
For anyA⊂N, define the positive upper density of A by(8)d¯A=lim supn⟶∞1nA∩0,1,…,n−1.
Define the lower density ofA by(9)d¯A=lim infn⟶∞1nA∩0,1,…,n−1.
IfA⊂N, Ac is the complementary set of A.
ω in the pseudo-orbit of the shadowing property,d¯-shadowing property, and d¯-shadowing property for IFSf0,f1 is the same as the one chosen in the shadowing orbit, while the two ωs in the definitions of the average shadowing property and asymptotic average shadowing property may be different.


## 2. Thed¯-Shadowing Property and d¯-Shadowing Property for IFSf0,f1

Bahabadi [10] introduced the definition of the shadowing property for IFSf0,f1. In this paper, we will introduce the definitions of the d¯-shadowing property and d¯-shadowing property for IFSf0,f1 and give some results.Definition 2.
A sequenceξii≥0 is called a δ-pseudo-orbit for IFSf0,f1 if there is ω∈Σ2 such that for any i∈N,(10)dfωiξi,ξi+1<δ.Definition 3.
IFS
f
0
,
f
1 has the shadowing property if for any ε>0, there is δ>0 such that every δ-pseudo-orbit ξii≥0 is ε-shadowed by a point z∈X, i.e., there is z∈X such that for any i∈N,(11)dfωiz,ξi<ε.Example 1.
Suppose thatf0x=0,f1x=1,x∈0,1. For any ε>0, there is δ=ε>0. Let ξii≥0 be a δ-ergodic-pseudo-orbit for IFSf0,f1. Therefore, there is ω=ω0ω1ω2⋯∈Σ2 such that(12)limn⟶∞1ni∈N:0≤i<n,dfωiξi,ξi+1<δ=1.
Then, there isz=ξ0 such that(13)limn⟶∞1ni∈N:0≤i<n,dfωiz,ξi<ε=1.
Obviously,IFSf0,f1 has the d¯-shadowing property and d¯-shadowing property.Lemma 1.
For anyk∈Z+, and any ε>0, there is 0<δ<ε/k such that there is any δ-chain of length k+1 for IFSf0,f1ξ0,ξ1,…,ξk, i.e., there is ω∈Σ2 such that dfωiξi,ξi+1<δ satisfies(14)dfωiξ0,ξi<ε,∀0≤i≤k.Proof.
Fixk∈Z+ and let ε>0. Since f0 and f1 are uniformly continuous, for any ω1=ω01ω11⋯∈Σ2, fω1i is also uniformly continuous, where i=0,1,…,k. Then for ε/k>0, there is 0<δ<ε/k such that dfω1ix,fω1iy<ε/k for any x,y∈X satisfies dx,y<δ. Since ξ0,ξ1,…,ξk is a δ-chain of IFSf0,f1, there is ω∈Σ2 such that(15)dfωiξi,ξi+1<δ,≤i≤k−1.
Putω1=ω. Then,(16)dfωjξ0,ξj≤dfωj−1⋯fω1fω0ξ0,fωj−1⋯fω1ξ1+⋯+dfωj−1ξj−1,ξj<εk+εk+⋯+εk+δ<k⋅εk=ε,where 0≤j≤k.Theorem 1.
(1)
IfIFSf0,f1 has the d¯-shadowing property, then ℱk has the d¯-shadowing property for any k∈Z+
(2)
IfIFSf0,f1 has the d¯-shadowing property, then ℱk has the d¯-shadowing property for any k∈Z+Proof.
Fixk∈Z+ and let ε>0. We can find a number δ>0 satisfying Lemma 1. We will show that 1 is true. We can find a number δ1>0 such that each δ1-ergodic-pseudo-orbit of IFSf0,f1 is δ-shadowed by a true orbit along a set with positive lower density, and δ>δ1. Let xii≥0 be a δ1-ergodic-pseudo-orbit of ℱk. That is to say, there is ω∈Σ2 such that(17)limn⟶∞1ni∈N:0≤i<n,dfωi+1k−1∘⋯∘fωikxi,xi+1<δ1=1.
Let(18)ξii≥0=x0,fω0x0,…,fωk−2∘⋯∘fω0x0,x1,fωkx1,…,fω2k−2∘⋯∘fωkx1,…,xn,fωnkxn,…,fωn+1k−2∘⋯∘fωnkxn,….
Obviously,ξii≥0 is a δ1-ergodic-pseudo-orbit of IFSf0,f1. Then, there is z∈X such that(19)lim infn⟶∞1ni∈N:0≤i<n,dfωiz,ξi<δ>0.
PutA=i∈N:dfωkiz,xi<ε,A1=i∈N:dfωiz,ξi<δ,B1=i∈N:dfωi+1k−1∘⋯∘fωikxi,xi+1<δ1,andM=i=kl+j:l∈B1,j=1,⋯,k.
Obviously,d¯A1>0,dM=1,dB1=1, and then d¯A1∩M=d¯A1>0. Now, to prove the theorem, we need to show the following claim.Claim 1.
PutA∩0,1,…,n−1/n=sn/nandA1∩M∩0,1,…,nk−1−k/nk=tn/nk, then sn/n≥tn/nk.Proof.
Without loss of generality, suppose thatdz,x0<ε, then the claim is true when n=1. Assume sn/n≥tn/nk, then we will show that sn+1/n+1≥tn+1/n+1k. For any m=kp+q∈A1∩M∩0,…,nk−1, where p∈N,0<q≤k, we have dfωmz,ξm<δ, and the sequence fωm−1z,ξm,⋯,ξkp+1−1,xp+1 is a δ-chain. According to Lemma 1, dfωkp+1z,xp+1<ε. So, p+1∈sn+1. Therefore, sn+1≥A1∩M∩0,1,…,nk/k and sn+1/n+1≥A1∩M∩0,1,⋯,n+1k−1−k/n+1k=tn+1/n+1k
Obviously,(20)lim infn⟶∞A1∩M∩0,1,…,nk−1−knk=lim infn⟶∞A1∩M∩0,1,…,nk−1nk.
Then,d¯A≥d¯A1∩M>0. So 1 is true. Now suppose that IFSf0,f1 has the d¯-shadowing property. In this case, the proof is the same as for 1 except that d¯A1∩M=d¯A1>1/2. Then, we can see that d¯A>1/2.Theorem 2.
(1)
Ifℱk has the d¯-shadowing property for some k∈Z+, then IFSf0,f1 has the d¯-shadowing property
(2)
Ifℱk has the d¯-shadowing property for some k∈Z+, then IFSf0,f1 has the d¯-shadowing propertyProof.
Suppose thatℱk has the d¯-shadowing property for some k∈Z+. Let ε>0, and we can find a number ε1>0 satisfying Lemma 1. Since ℱk has the d¯-shadowing property, there is δ1>0 such that any δ1-ergodic-pseudo-orbit of ℱk is ε1-shadowed by a true orbit along a set with positive lower density, and ε1>δ1. For δ1, we can get δ>0 by Lemma 1. Let ξii≥0 be a δ-ergodic-pseudo-orbit of IFSf0,f1, and xii≥0=ξkii≥0. Therefore, there is ω∈Σ2 such that(21)limn⟶∞1ni∈N:0≤i<n,dfωiξi,ξi+1<δ=1.
LetM=i∈N:0≤i<n,dfωik+jξik+j,ξik+j+1<δ,j=0,…,k−1, then(22)limn⟶∞1nM=1.
Therefore, ifi∈M, then the sequence ξik,ξik+1,…,ξi+1k is a δ-chain. According to Lemma 1, dfωi+1k−1∘⋯∘fωikxi,xi+1=dfωi+1k−1∘⋯∘fωikξik,ξi+1k<δ1, so xii≥0 is a δ1-ergodic-pseudo-orbit of ℱk. There is z∈X such that(23)lim infn⟶∞1ni∈N:0≤i<n,dfωkiz,xi<ε1>0.
PutA=i∈N:dfωkiz,xi<ε1 and A1=i∈N:dfωiz,ξi<ε, then d¯A∩M=d¯A>0. There is m∈A∩M such that dfωkmz,ξkm=dfωkmz,xm<ε1, and then the sequence fωkm−1z,xm,ξkm+1,⋯,ξkm+k−1 is a ε1-chain. According to Lemma 1, we have dfωkm+jz,ξkm+j<ε, where j=0,…,k−1. Now, to prove the theorem, we need to show the following claim.Claim 2.
PutA∩M∩0,1,…,n−1/n=sn/nandA1∩0,1,…,nk−1/nk=tn/nk, then tn/nk≥sn/n.Proof.
Without loss of generality, suppose that1∈M and dz,x0<ε1, then the claim is true when n=1. Assume that tn/nk≥sn/n, then we will show that tn+1/n+1k≥sn+1/n+1. Obviously, sn+1=sn or sn+1=sn+1. If sn+1=sn+1, then dfωknz,xn<ε1, and we can see that dfωkn+jz,ξkn+j<ε, where j=0,…,k−1. So tn+1=tn+k and tn+1/n+1k≥sn+1/n+1acs. If sn+1=sn, then tn+1≥tn and tn+1/n+1k≥sn+1/n+1.
Then,d¯A1≥d¯A∩M>0. So 1 is true. Now suppose that ℱk has the d¯-shadowing property for some k∈Z+. In this case, the proof is the same as for 1 except that d¯A∩M=d¯A>1/2. Then, we can see that d¯A1>1/2.Corollary 1.
LetIFSf0,f1 be a iterated function system, then the following statements are equivalent:(1)
IFSf0,f1 has the d¯-shadowing property (respectively, d¯-shadowing property)(2)
ℱk has the d¯-shadowing property (respectively, d¯-shadowing property) for some k∈Z+(3)
ℱk has the d¯-shadowing property (respectively, d¯-shadowing property) for any k∈Z+Lemma 2.
LetA,B⊂N; 1 if d¯A=α and d¯B>1−α, then d¯A∩B>0; 2 if d¯A=α and d¯B>1−α, then d¯A∩B>0. Here, 0≤α≤1.Proof.
Firstly, we will show1. Suppose on the contrary that d¯A∩B=0. Then, d¯B≤d¯Ac. Therefore, we can see d¯A+d¯Ac≥d¯A+d¯B>α+1−α=1. But d¯A+d¯Ac=1, which contradicts the hypothesis. Hence, d¯A∩B>0. The proof for the second case is the same as for 1.Theorem 3.
f
0 or f1 is surjective,(1)
IfIFSf0,f1 has the d¯-shadowing property, then IFSf0,f1 is chain transitive(2)
IfIFSf0,f1 has the d¯-shadowing property, then IFSf0,f1 is chain transitiveProof.
Without loss of generality, suppose thatf1 is surjective. Then, there is a sequence yii≥0=y0=y,y1,y2,⋯ such that for any i>0, f1yi=yi−1. Let ε>0 be arbitrary, and x,y∈X. Firstly, we will prove that (1) is true. We can find a number δ>0 as in the definition of the d¯-shadowing property for IFSf0,f1. Put a1=2,ai=2a1+⋯+ai−1. Construct a sequence as follows: ξii≥0=x,f0x,f0a1x,ya2,ya2−1,⋯,y,x,f0x,⋯,f0a3x,ya4,⋯,y,⋯. Then, ξii≥0 is a δ-ergodic-pseudo-orbit for IFSf0,f1, i.e., there is ω=ω0ω1ω2⋯∈Σ2 such that(24)limn⟶∞1ni∈N:0≤i<n,dfωiξi,ξi+1<δ=1,where(25)ωi=0,df0ξi,ξi+1<δ,1,df1ξi,ξi+1≥δ.
PutA1=i∈N:ξi∈f0jxj≥0andA2=i∈N:ξi∈f1jyj≥0. Then, d¯A1=1,d¯A2=1. Since IFSf0,f1 has the d¯-shadowing property, there is z∈X such that(26)lim infn⟶∞1ni∈N:0≤i<n,dfωiz,ξi<ε>0.
According to Lemma2, there are i0,j0,l,s∈N,l<s such that(27)dfωlz,ξl<ε,dfωsz,ξs<ε,ξl=f0i0x,ξs=yj0.
Sox,f0x,f02x,⋯,f0i0−1x,fωlz,⋯,fωs−1z,yj0,yj0−1,⋯,y1,y is an ε-chain from x to y. Hence, IFSf0,f1 is chain transitive.
Then, we will prove that (2) is true. Construct a sequence as follows:(28)ξii≥0=x,y,x,f0x,y1,y,x,f0x,f02x,y2,y1,y,⋯.
Then,ξii≥0 is a δ-ergodic-pseudo-orbit for IFSf0,f1, i.e., there is ω=ω0ω1ω2⋯∈Σ2 such that(29)limn⟶∞1ni∈N:0≤i<n,dfωiξi,ξi+1<δ=1,where(30)ωi=0,df0ξi,ξi+1<δ,1,df1ξi,ξi+1≥δ.
Then,d¯A1=1/2,d¯A2=1/2. Since IFSf0,f1 has the d¯-shadowing property, there is z∈X such that(31)lim supn⟶∞1ni∈N:0≤i<n,dfωiz,ξi<ε>12.
Then, the proof of this case is the same as for1.Theorem 4.
IfIFSf0,f1 has the d¯-shadowing property or d¯-shadowing property, and f0 or f1 is surjective, then IFSf0,f1 is chain mixing.Proof.
The result is obtained by Corollary1, Theorem 3, and Theorem 2.3 in [13].Lemma 3 (see [10]).
IFS
f
0
,
f
1 is chain transitive and has the shadowing property, and then IFSf0,f1 is transitive.Corollary 2.
IFS
f
0
,
f
1 has the d¯-shadowing property (respectively, d¯-shadowing property) and the shadowing property, one of f0,f1 is surjective, and then IFSf0,f1 is mixing.Proof.
The result is obtained by Theorem4 and Lemma 3.
IFS
f
0
,
f
1 is ergodic if for any pair of nonempty open subsets U,V⊂X, there is ω∈Σ2 such that NU,V=n∈N:fωnU∩V≠∅ has the positive upper density. IFSf0,f1 is strongly ergodic if for any pair of nonempty open subsets U,V⊂X, there is ω∈Σ2 such that NU,V=n∈N:fωnU∩V≠∅ is a syndetic set.
A mapf:X⟶X is an open map if for any open set U⊂X, fU is also an open set.Theorem 5.
Letf0,f1 be open maps. For IFSf0,f1 with the d¯-shadowing property (respectively, d¯-shadowing property), if A⊂X is dense in X, and s is a minimal point of f0 or f1 for any s∈A, then IFSf0,f1 is strongly ergodic.Proof.
Suppose thatU and V are two nonempty open subsets of X, then there are x∈U∩A,y∈V∩A,andε>0 such that Bx,ε⊂U,By,ε⊂V and Jxp,Jyq are syndetic sets, where Jxp=i∈N:fpix∈Bx,ε/2,Jyq=i∈N:fqiy∈By,ε/2,p,q=0or1. Since Jxp,Jyq are syndetic sets, we can find Nx,Ny∈Z+ such that Jxp∩n,n+Nx≠∅,Jyq∩n,n+Ny≠∅ for any n∈N. Firstly, we will prove if IFSf0,f1 has the d¯-shadowing property, then the conclusion is true. Put N0=maxNx,Ny,a0=0,a1=N0,a2=2a0+a1,…,an=2a0+a1+a2+⋯+an−1,andEn=a0+a1+⋯+an. Put(32)E=∪n=0+∞E2n,E2n+1,…,E2n+1−1,F=∪n=0+∞E2n+1,E2n+1+1,…,E2n+2−1.
We can see thatd¯E=1andd¯F=1. For ε/2>0, we can find a number ε1>0 satisfying Lemma 1, where k=N0+1. There is δ>0 such that every δ-ergodic-pseudo-orbit of IFSf0,f1 is ε1-shadowed by a true orbit along a set with positive lower density, and ε1>δ. Put ξii≥0 as follows:(33)ξi=fpi−E2nx,E2n≤i<E2n+1,fqi−E2n+1y,E2n+1≤i<E2n+2.
That is to say,ξii≥0=x,fpx,⋯,fpa1−1x,y,fqy,⋯,fqa2−1y,x,⋯,fpa3−1x,⋯. Then, ξii≥0 is a δ-ergodic-pseudo-orbit for IFSf0,f1, i.e., there is ω=ω0ω1ω2⋯∈Σ2 such that(34)limn⟶∞1ni∈N:0≤i<n,dfωiξi,ξi+1<δ=1,where(35)ωi=p,ξi=fpi−E2nx,q,ξi=fqi−E2n+1y.
SinceIFSf0,f1 has the d¯-shadowing property, there is z∈X such that(36)lim infn⟶∞1ni∈N:0≤i<n,dfωiz,ξi<ε1>0.
LetB=0≤i<n:dfωiz,ξi<ε1, then d¯B>0. According to Lemma 2, both B∩EandB∩F are infinite. Therefore, we can find n0∈B∩Eandm0∈B∩F such that E2t0+1−n0≥2N0andE2s0+2−m0≥2N0, where t0,s0∈N,t0<s0. Suppose on the contrary that for any n0∈B∩E, E2t+1−n0<2N0, where t∈N. We can see(37)d¯B=lim infn⟶∞1ni∈N:0≤i<n,dfωiz,ξi<ε1≤limt⟶∞1E2t+1B∩0,1,…,E2t+1−1≤limt⟶∞a0+2N0+a2+2N0+a4+⋯+a2t+2N0E2t+1≤limt⟶∞a0+a1+⋯+a2t+2N0E2t+1=limt⟶∞a0+a1+⋯+a2t+2N0a0+a1+⋯+a2t+a2t+1=limt⟶∞a0+a1+⋯+a2t+2N0a0+a1+⋯+a2t+2a0+a1+⋯+a2t=0,which is in contrast with d¯B>0. Another proof can be obtained similarly. Since n0∈B∩E and m0∈B∩F, dfωn0z,ξn0<ε1 and dfωm0z,ξm0<ε1, where ξn0=fpn0−E2t0x and ξm0=fqm0−E2s0+1y. Owing to x,y∈A, there are 0≤n∗ and m∗≤N0 such that dfpn0−E2t0+n∗x,x<ε/2 and dfqm0−E2s0+1+m∗y,y<ε/2. Obviously, fωn0−1z,ξn0,ξn0+1,⋯,ξn0+n∗,⋯,ξn0+N0 and fωm0−1z,ξm0,ξm0+1,⋯,ξm0+m∗,⋯,ξm0+N0 are ε1-chains. According to Lemma 1, dfωn0+n∗z,fpn0−E2t0+n∗x<ε/2anddfωm0+m∗z,fqm0−E2s0+1+m∗y<ε/2. Therefore, dfωn0+n∗z,x<εanddfωm0+m∗z,y<ε. Then, fωn0+n∗z∈Bx,ε⊂U and fωm0+m∗z∈By,ε⊂V. Obviously, m0+m∗>n0+n∗. Put N∗=m0+m∗−n0+n∗ and ω1=ω01ω11⋯ωN∗−11⋯=ωn0+n∗⋯ωm0+m∗−1⋯, then fω1N∗fωn0+n∗z=fωm0+m∗z∈By,ε⊂V. Hence, fω1N∗U∩V≠∅.
We writeW=fω1N∗U∩V≠∅. As f0,f1 are open sets, fω1N∗U is an open set. Therefore, W is an open set. Then, there are z∗∈W∩A and ω2∈Σ2 such that J=n∈N:fω2nz∗∈W is a syndetic set. For any m∈J, fω2mW∩W≠∅. There is ω3=ω03ω13⋯=ω01ω11⋯ωN∗−11ω2∈Σ2 such that ∅≠fω2mW∩W=fω2mfω1N∗U∩V∩fω1N∗U∩V⊂fω3N∗+mU∩V. Since N∗+m:m∈J⊂NU,V=n∈N:fω3N∗+mU∩V≠∅, NU,V is a syndetic set. As U and V are arbitrary, IFSf0,f1 is strongly ergodic.
Then, we will prove ifIFSf0,f1 has the d¯-shadowing property, then the conclusion is true. Put a0=0,a1=N0,a2=2⋅N0,…,an=n⋅N0,andEn=a0+a1+⋯+an.(38)E=∪n=0+∞E2n,E2n+1,⋯,E2n+1−1,F=∪n=0+∞E2n+1,E2n+1+1,⋯,E2n+2−1.
We can see thatd¯E=1/2andd¯F=1/2. Then, the proof of this case is the same as for case one.Theorem 6.
Letf0,f1 be open maps. For IFSf0,f1 with the d¯-shadowing property (respectively, d¯-shadowing property), if A⊂X is dense in X, and s is a minimal point of f0 or f1 for any s∈A, then ℱk is strongly ergodic.Proof.
According to the condition, for anys∈S, s is a minimal point of f0 or f1. For any k∈Z+, it is well known that APf=APfk and then s is a minimal point of f0k or f1k. We can combine Corollary 1 and Theorem 5. Then, ℱk is strongly ergodic.
A pointx∈X is called a stable point of IFSf0,f1 if for any ε>0, there is δ>0 satisfying dfωnx,fωny<ε for any y∈X with dx,y<δ and any ω∈Σ2,n∈N. IFSf0,f1 is called Lyapunov stable if any point of X is a stable point of IFSf0,f1.Theorem 7.
ForIFSf0,f1 with the d¯-shadowing property (respectively, d¯-shadowing property), if IFSf0,f1 is Lyapunov stable, and f0 or f1 is surjective, then IFSf0,f1 is transitive. Hence, ℱk is transitive for any k∈Z+.Proof.
Suppose thatU and V are two nonempty open subsets of X, then there are x∈U,y∈V,andε>0 such that Bx,ε⊂UandBy,ε⊂V, where x,y∈X are stable points of IFSf0,f1. There is ε1>0 such that for any u,v∈X,(39)du,v<ε1,thendfωnu,fωnv<ε2,for  alln=0,1,2….
Without loss of generality, suppose thatf1 is surjective. Then, there is a sequence y−jj≥0 such that y−j+1=fy−j for all j∈Z+ and y0=y. Firstly, we will prove if IFSf0,f1 has the d¯-shadowing property, then the conclusion is true. Put a0=0,a1=1,a2=2a0+a1,…,an=2a0+a1+a2+⋯+an−1,anden=a0+a1+⋯+an Put(40)E=∪n=0+∞e2n,e2n+1,⋯,e2n+1−1,F=∪n=0+∞e2n+1,e2n+1+1,⋯,e2n+2−1,then we can see that d¯E=1andd¯F=1. For ε1>0, we can find a number δ>0 as in the definition of the d¯-shadowing property for IFSf0,f1. Construct a sequence ξii≥0 as follows:(41)ξi=f0i−e2nx,e2n≤i<e2n+1,yi−e2n+2,e2n+1≤i<e2n+2.
That is to say,ξii≥0=x,y−1,x,⋯,f0a3−1x,y−a4,⋯y−1,⋯. Then, ξii≥0 is a δ-ergodic-pseudo-orbit for IFSf0,f1, i.e., there is ω1=ω01ω11ω21⋯∈Σ2 such that(42)limn⟶∞1ni∈N:0≤i<n,dfωi1ξi,ξi+1<δ=1,where(43)ωi1=0,ξi=f0i−e2nx,1,ξi=yi−e2n+2.
SinceIFSf0,f1 has the d¯-shadowing property, there is z0∈X such that(44)lim infn⟶∞1ni∈N:0≤i<n,dfω1iz0,ξi<ε1>0.
LetB=0≤i<n:dfω1iz0,ξi<ε1, then d¯B>0. According to Lemma 2, both B∩EandB∩F are infinite. Therefore, there are n0∈B∩Eandm0∈B∩F such that e2t0≤n0<e2t0+1ande2s0+1≤m0<e2s0+2, where t0,s0∈N,t0<s0; then, dfω1n0z0,ξn0<ε1 and dfω1m0z0,ξm0<ε1, where ξn0=f0n0−e2t0x and ξm0=ym0−e2s0+2. According to 2.1, dfω1e2s0+2z0,y=dfωe2s0+2−11∘⋯∘fωm01fω1m0z0,fωe2s0+2−11∘⋯∘fωm01ym0−e2s0+2<ε/2 and dfωe2s0+2−11∘⋯∘fωn01fω1n0z0,fωe2s0+2−11∘⋯∘fωn01f0n0−e2t0x=dfω1e2s0+2z0,fωe2s0+2−11∘⋯∘fωn01f0n0−e2t0x=dfω1e2s0+2z0,fωe2s0+2−11∘⋯∘fωn01fωn0−11∘⋯∘fωe2t01x<ε/2.
Then,dy,fωe2s0+2−11∘⋯∘fωn01fωn0−11∘⋯∘fωe2t01x<ε. Put ω2=ω02ω12⋯=ωe2t01ωe2t0+11⋯∈Σ2 and N0=e2s0+2−e2t0, then fω2N0U∩V≠∅. Hence, IFSf0,f1 is transitive. According to Corollary 1, ℱk is transitive for any k∈Z+.
Then, we will prove ifIFSf0,f1 has the d¯-shadowing property, then the conclusion is true. Put a0=0,a1=1,a2=2,…,an=n,andEn=a0+a1+⋯+an.(45)E=∪n=0+∞E2n,E2n+1,…,E2n+1−1,F=∪n=0+∞E2n+1,E2n+1+1,…,E2n+2−1.
We can see thatd¯E=1/2andd¯F=1/2. Then, the proof of this case is the same as for case one.


## 3. The Average Shadowing Property forIFSf0,f1

Bahabadi [10] introduced the definition of the average shadowing property for IFSf0,f1. In this section, we will research the relationship among the average shadowing property, ergodicity, and strong ergodicity.Definition 4.
A sequenceξii≥0 is called a δ-average-pseudo-orbit for IFSf0,f1 if there are ω∈Σ2 and N∈Z+ such that for every n≥N,(46)1n∑i=0n−1dfωiξi,ξi+1<δ.Definition 5.
IFS
f
0
,
f
1 has the average shadowing property ASP if for any ε>0, there is δ>0 such that every δ-average-pseudo-orbit ξii≥0 is ε-shadowed on average by a point z∈X, i.e., there is ω∈Σ2 such that(47)limsupn⟶∞1n∑i=0n−1dfωiz,ξi<ε.Theorem 8.
ForIFSf0,f1 with the ASP, if S⊂X is dense in X, and s is a quasi-weakly almost periodic point of f0 or f1 for any s∈S, then IFSf0,f1 is ergodic.Proof.
Suppose thatU,V⊂X are two nonempty open sets. We can find two points u∈U, v∈V, and ε>0 such that Bu,ε⊂UandBv,ε⊂V. There is ε0>0 such that x∈Bx,ε0⊂Bu,ε/2,y∈By,ε0⊂Bv,ε/2, where x,y∈S. Therefore, Jxp=i∈N:fpix∈Bu,ε/2 and Jyq=i∈N:fqiy∈Bv,ε/2 have the positive upper density, where p,q=0or1. We can find k1,k2∈N such that d¯Jxp=α>4/k1≥4/k>0,d¯Jyq=β>4/k2≥4/k>0,wherek=maxk1,k2. Since IFSf0,f1 has the ASP, there is δ>0 such that every δ-average-pseudo-orbit of IFSf0,f1 is ϵ/2k-shadowed on average by a point in X. We can find a number N0∈N such that 3D/N0<δ, and N0=t⋅k, where D=diamX, t∈Z+. Define the sequence ξii≥0 as follows: ξii≥0=x,fpx,⋯,fpN0−1x,y,fqy,⋯,fqN0−1y,fpN0x,⋯, fp2N0−1x,fqN0y,⋯.
We can choose(48)ωi=p,dfpξi,ξi+1<δ,q,dfpξi,ξi+1≥δ.
ω
=
ω
0
ω
1
ω
2
⋯
∈
Σ
2. Obviously, for n≥N0,(49)1n∑i=0n−1dfωiξi,ξi+1≤n/N0⋅3Dn≤3DN0<δ.
Soξii≥0 is a δ-average-pseudo-orbit for IFSf0,f1. Hence, there are z∈Xandω1∈Σ2 such that(50)lim supn⟶∞1n∑i=0n−1dfω1iz,ξi<ε2k.
Sinced¯Jxp=α>4/k1≥4/k>0,d¯Jyq=β>4/k2≥4/k>0,k=maxk1,k2, we have d¯Wxp>2/k,d¯Wyq>2/k, where Wxp=i∈N:ξi∈Bu,ε/2, Wyq=i∈N:ξi∈Bv,ε/2. Put F=i∈N:dfω1iz,ξi≥ε/2. Then, d¯F<1/k. Suppose on the contrary that d¯F≥1/k, then(51)lim supn⟶∞1n∑i=0n−1dfω1iz,ξi≥lim supn⟶∞1nF∩0,1,…,n−1⋅ε2+Fc∩0,1,…,n−1⋅0≥1k⋅ε2=ε2k,which is in contrast with (50). So d¯F<1/k. Therefore, d¯Fc=1−d¯F≥1−1/k=k−1/k. According to Lemma 2, d¯Wxp∩Fc>0 and d¯Wyq∩Fc>0. There are i0,j0,l,s,s>l such that fpi0x∈Bu,ε/2,fqj0y∈Bv,ε/2, and dfω1lz,ξl<ε/2, dfω1sz,ξs<ε/2, where ξl=fpi0x,ξs=fqj0y. Then, dfω1lz,u<ε,dfω1sz,v<ε. Therefore, fω1lz∈Bu,ε⊂U and fωs−11∘⋯∘fωl1fω1lz=fω1sz∈Bv,ε⊂V. Let n0=s−l, and ω2=ω02ω12⋯=ωl1ωl+11⋯ωs−11⋯∈Σ2, then NU,V=n0∈Z+:fω2n0U∩V,ω2∈Σ2≠∅. Obviously,(52)lim supn⟶∞n0:n0=s−l,s>l,l∈Wxp∩Fc,s∈Wyq∩Fc∩0,…,n−1n>0.
Therefore,d¯NU,V>0. As UandV are arbitrary, IFSf0,f1 is ergodic.Theorem 9.
Letf0,f1 be open maps. For IFSf0,f1 with the ASP, if S⊂X is dense in X, and s is a minimal point of f0 or f1 for any s∈S, then IFSf0,f1 is strongly ergodic.Proof.
Suppose thatU and V are two nonempty open subsets of X. Obviously, the syndetic set has the positive upper density. By Theorem 8, there are n0∈N,ω2=ω02ω12⋯ωn0−12⋯∈Σ2 such that fω2n0U∩V≠∅. We write W=fω2n0U∩V≠∅. As f0,f1 are open sets, fω2n0U is an open set. Therefore, W is an open set. Then, there are z∗∈W∩S, ω3∈Σ2 such that J=n∈N:fω3nz∗∈W is a syndetic set. If m∈J, then fω3mW∩W≠∅. There is ω4=ω04ω14⋯=ω02ω12⋯ωn0−12ω3∈Σ2 such that ∅≠fω3mW∩W=fω3mfω2n0U∩V∩fω2n0U∩V⊂fω4n0+mU∩V. Since n0+m:m∈J⊂NU,V=n∈N:fω4n0+mU∩V≠∅, NU,V is a syndetic set. As U and V are arbitrary, IFSf0,f1 is strongly ergodic.Lemma 4 (Theorem 3.1 in [13]).
IfIFSf0,f1 has the ASP, then ℱk has the ASP for any k∈Z+.Theorem 10.
Letf0,f1 be open maps. For IFSf0,f1 with the ASP, if S⊂X is dense in X, and s is a minimal point of f0 or f1 for any s∈S, then ℱk is strongly ergodic.Proof.
According to the condition, for anys∈S, s is a minimal point of f0 or f1. For any k∈Z+, it is well known that APf=APfk and then s is a minimal point of f0k or f1k. We can combine Lemma 4 and Theorem 9. Then, ℱk is strongly ergodic.


## 4. A Remark on the Asymptotic Average Shadowing Property forIFSf0,f1

The definition of the asymptotic average shadowing property forIFSf0,f1 is introduced by Nia [11]. Here, we will come up with a situation to show that an example does not have the asymptotic average shadowing property.Definition 6.
A sequenceξii≥0 is called an asymptotic average pseudo-orbit for IFSf0,f1 if there is ω∈Σ2 such that(53)limn⟶∞1n∑i=0n−1dfωiξi,ξi+1=0.Definition 7.
IFS
f
0
,
f
1 has the asymptotic average shadowing property AASP if every asymptotic average pseudo-orbit is shadowed on average by a point z∈X, i.e., there is ω∈Σ2 such that(54)limn⟶∞1n∑i=0n−1dfωiz,ξi=0.
For anyA,B⊂X,(55)dA,B=mina∈A,b∈Bda,b.Theorem 11.
IfU⊂X is an open set, then U is invariant under ftt=0,1ftU⊂U,t=0,1. Assume that there is x∈X such that f0ix¯i≥0⊂U or f1ix¯i≥0⊂U. Suppose that y∈X is a point whose orbit is metrically separated from U (for any ω∈∑2, dfωjyj≥0,U>0). Then, IFSf0,f1 does not have the AASP.Proof.
Without loss of generality, supposef0ix¯i≥0⊂U. Construct ξii≥0 as follows: ξii≥0=x,y,x,f0x,y,f1y,x,f0x,f02x,f03x,y,f1y,f12y,f13y,x,f0x,⋯, f07x,y,….
For anyk∈N, we can find l∈N with(56)∑i=1l2i≤k≤∑i=1l+12i.
We can choose(57)ωi=0,df0ξi,ξi+1=0,1,df0ξi,ξi+1>0.
ω
=
ω
0
ω
1
ω
2
⋯
∈
Σ
2. We can see that(58)∑i=0k−1dfωiξi,ξi+1≤3l⋅D,D=diamX.
This implies that(59)limn⟶∞1n∑i=0n−1dfωiξi,ξi+1≤liml⟶∞3l⋅D2l=0.
Soξii≥0 is an asymptotic average pseudo-orbit of IFSf0,f1. We will prove that IFSf0,f1 does not have the AASP. Suppose that IFSf0,f1 has the AASP. For asymptotic average pseudo-orbit ξii≥0, there are z∈X,ω1∈Σ2 such that ξii≥0 is asymptotically shadowed on average by the point z. Meanwhile, the orbit of z has to enter U at some point. Otherwise,(60)12n∑i=02n−1dfω1iz,ξi≥12dfω1jzj≥0,U>0,∀n∈N.
SoIFSf0,f1 does not have the AASP. Therefore, there is N0∈N such that fω1N0z∈U. Then, fω1nz∈U for any n≥N0. Therefore, for n that is large enough,(61)12n∑i=02n−1dfω1iz,ξi≥14df1jyj≥0,U>0,which contradicts with the hypothesis. IFSf0,f1 does not have the AASP.
The following example comes from the study in [13]. It is a vitally important example of IFSf0,f1. It is controversial whether it has the AASP. We can use the above theorem to prove that it does not have the AASP.Example 2.
Letf0,f1: 0,1⟶0,1 be two continuous maps such that f1x>f0x>x if and only if x∈0,1/2 and f00=f10=1/4. Obviously, we can see that f0i1/8¯i≥0⊂U=0,1/2, ftU⊂U, where t=0,1. And for any ω∈∑2, dfωj7/8j≥0,U>0. According to Theorem 11, IFSf0,f1 does not have the AASP.



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