In this article we consider iterated function systems that contain inverse maps, which we call IFS with inverses. We show that the invariant measures for IFS with inverses agree with the invariant measures for associated graph-directed IFS under the suitable choice of weight.
Citation: Yuki Takahashi. On iterated function systems with inverses[J]. AIMS Mathematics, 2025, 10(4): 9034-9041. doi: 10.3934/math.2025415
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In this article we consider iterated function systems that contain inverse maps, which we call IFS with inverses. We show that the invariant measures for IFS with inverses agree with the invariant measures for associated graph-directed IFS under the suitable choice of weight.
A finite collection of strictly contractive maps on the real line is called an iterated function system (IFS). Let Φ={φa}a∈Λ be an IFS and p=(pa)a∈Λ be a probability vector. It is well-known that there exists a unique Borel probability measure ν, called the invariant measure, such that
ν=∑a∈Λpa⋅φaν, |
where φaν is the push-forward of ν under the map φa:R→R.
When the construction does not involve complicated overlaps (say, under the strong separation condition), the invariant measures are relatively easy to understand. For example, if the strong separation condition holds, then the invariant measure ν is supported on a Cantor set and is singular, and the dimension of ν is given by
dimν=hχ, |
where h=h(p) is the entropy and χ=χ(Φ,p) is the Lyapunov exponent.
In this paper we consider IFS with inverses (i.e., IFS that contain inverse maps). IFS with inverses were first introduced by the author in [5], motivated by the Furstenberg measure. See also [6]. We show that the invariant measures for IFS with inverses agree with the invariant measures for associated graph-directed IFS under the suitable choice of weight. The main results of [5] and [6] follow directly from our result.
The paper is organized as follows: In section 2, we recall IFS with inverses and state the main result. Section 3 is devoted to preliminary lemmas. In section 4 we prove the main result.
Let G be the free group of rank r≥2, and let W be a free generating set of G. Let Λ be a set that satisfies
W⊂Λ⊂W∪W−1, |
where W−1={a−1}a∈W. Let E∗=⋃n≥1Λn and E=ΛN. For ω=ω0ω1⋯ we denote ω|n=ω0⋯ωn−1. For ω,ξ∈E∪E∗ we denote by ω∧ξ their common initial segment. For ω∈E∗ and ξ∈E∪E∗, we say that ω precedes ξ if ω∧ξ=ω.
Let p=(pa)a∈Λ be a non-degenerate probability vector, and let μ be the associated Bernoulli measure on E. We say that a (finite or infinite) sequence ω∈E∗∪E is reduced if ωiωi+1≠aa−1 for all i≥0 and a∈Λ. Let Γ∗ (resp. Γ) be the set of all finite (resp. infinite) reduced sequences. For ω∈Γ∗ we denote the associated cylinder set in Γ by [ω]. Define the map
red:E∗→Γ∗ |
in the obvious way, i.e., red(ω) is the sequence derived from ω by deleting all occurrences of consecutive pairs aa−1 (a∈Λ). Let ¯E⊂E be the set of all ω such that the limit
limn→∞red(ω|n) | (1) |
exists. For example, for any a∈Λ we have aaa⋯∈¯E and aa−1aa−1⋯∉¯E. By abuse of notation, for ω∈¯E we denote the limit (1) by red(ω). The following is well-known (see, e.g., chapter 14 in [3]):
Lemma 2.1. There exists 0<ℓ≤1 (drift or speed) such that
limn→∞1n|red(ω|n)|=ℓ |
for μ-a.e. ω∈E. In particular, ¯E has full measure.
Denote
Λ⋆={(a,b)∈Λ2:a≠b−1}. |
For a∈Λ, write Ra=R×{a}. We freely identify Ra with R below. Let Xa⊂Ra (a∈Λ) be open intervals and write X=⋃a∈ΛXa. Assume that there exist 0<γ<1 and 0<θ≤1 such that for all (a,b)∈Λ⋆, the map φab:Xb→Xa is C1+θ and satisfies
(ⅰ) ¯φab(Xb)⊂Xa;
(ⅱ) 0<|φ′ab(x)|<γ for all x∈Xb;
(ⅲ) φ−1ab:φab(Xb)→Xb is C1+θ.
We say that Φ={φab}(a,b)∈Λ⋆ is an IFS with inverses. For ω=ω0⋯ωn∈Γ∗, we denote
φω=φω0ω1∘⋯∘φωn−1ωn. |
Let Π:Γ→X be the natural projection map, i.e.,
Π(ω)=⋂n≥1φω|n+1(¯Xωn). |
Define ΠE:¯E→X by ΠE=Π∘red. Define the measure ν=ν(Φ,p) by ν=ΠEμ (i.e., the push-forward of the measure μ under the map ΠE:¯E→X). We call ν an invariant measure. It is easy to see that if Λ=W, then the measure ν is an invariant measure of an IFS. Let χ=χ(Φ,p) be the Lyapunov exponent, and hRW=hRW(p) be the random walk entropy. See section 3 in [5] for the precise definition. Fix xa∈Xa for each a∈Λ. For ω∈¯E and n∈N we denote xω,n=xj, where j=j(ω,n)∈Λ is the last letter of red(ω|n).
Proposition 2.1 (Proposition 3.1 in [5]). We have
χ=−limn→∞1nlog|φ′red(ω|n)(xω,n)| |
for μ-a.e. ω.
Notice that an IFS with inverses Φ={φab}(a,b)∈Λ⋆ does not have any explicit inverse map. The next example illustrates why we call Φ an IFS with inverses. For more detail, see Example 2.1 and Appendix in [5].
Example 2.1. Let r=2, W={0,1} and Λ={0,1,1−1}. For 0<k,l<1, define
f0(x)=kx, f1(x)=(1+l)x+1−l(1−l)x+1+l. |
Let f1−1=f−11. It is easy to see that we have f0(0)=0, f1(−1)=−1, f1(1)=1 and f′0(0)=k, f′1(1)=l. It is well-known that there exists a unique Borel probability measure ν that satisfies
ν=∑a∈Λpafaν. |
The above measure is called a Fustenberg measure. See., e.g., [2]. Let
Y0=(−k,k), Y1=(f1(−k),1) and Y−1=(−1,f−1(k)). |
Then we have
fa(Y∖Ya−1)⊂Ya, |
for all a∈Λ, where Y=⋃a∈ΛYa and Y0−1=∅. Notice that the sets {Ya}a∈Λ are not mutually disjoint if and only if k>f1(−k), which is equivalent to
√l>1−k1+k. |
It is easy to see that there exist open intervals X0,X1,X1−1⊂R such that
Ya⊂Xa and ¯fa(X∖Xa−1)⊂Xa |
for all a∈Λ, where X=⋃a∈ΛXa and X0−1=∅. Then {fa|Xb}(a,b)∈Λ⋆ is an IFS with inverses, and the associated invariant measure agrees with ν. For the proof, see the Appendix in [5].
Denote
dimν=inf{dimHY:ν(R∖Y)=0}. |
Proposition 2.2 (Proposition 3.3 in [5]). Assume that for all a∈Λ, the sets {¯φab(Xb)}b∈Λ⋆a are mutually disjoint, where
Λ⋆a={b∈Λ:(a,b)∈Λ⋆}. |
Then we have
dimνa=hRWχ |
for all a∈Λ.
Given an IFS with inverses Φ={φab}(a,b)∈Λ⋆, one can naturally associate a graph-directed IFS by restricting transitions from a to a−1 for all a∈W. For the precise definitions of graph-directed IFS, see section 1.7 in [1].
Let ˜P=(˜pab) be a |Λ|×|Λ| stochastic matrix that satisfies ˜pab>0 (a≠b−1) and ˜pab=0 (a=b−1). Let ˜p=(˜p1,⋯,˜pN) be the unique row vector satisfying ˜p˜P=˜p. Let ˜μ be the probability measure on Γ associated with ˜P and ˜p. Define the measure ˜ν=˜ν(Φ,˜P) by ˜ν=Π˜μ. For a∈Λ, denote ˜νa=˜ν|Xa. It is easy to see that
˜νa=∑(a,b)∈Λ⋆˜pab⋅φab˜νb. |
Let ˜h=˜h(˜P) be the entropy and ˜χ=˜χ(Φ,˜P) be the Lyapunov exponent, i.e.,
˜h=−∑(a,b)∈Λ⋆˜pa˜pablog˜pab, |
and
˜χ=−limn→∞1nlog|φ′ω|n(xωn)| |
for ˜μ-a.e. ω. Under the separation condition, we obtain the following. The argument is classical, so we omit the proof. See, e.g., the proof of (2.6) in [4].
Proposition 2.3. For every a∈Λ, assume that the sets {¯φab(Xb)}b∈Λ⋆a are mutually disjoint. Then we have
dim˜νa=˜h˜χ |
for all a∈Λ.
Our main result is the following:
Theorem 2.1. Let Φ={φab}(a,b)∈Λ⋆ be an IFS with inverses. Then there exists a stochastic matrix ˜P such that
ν=˜ν, hRW=ℓ˜h and χ=ℓ˜χ. |
Since the graph directed IFS has essentially the same structure as IFS, by the above theorem most of the results of IFS can be immediately extended to IFS with inverses. For example, the main results of [5] and [6] follow directly from the above result.
Define μred by redμ, i.e., the push-forward of the measure μ under the map red:E→Γ. From below, for n>0, which is not necessarily an integer, we interpret ω|n to be ω|⌊n⌋. The following lemma is immediate.
Lemma 3.1. We have
χ=−limn→∞1nlog|φ′ω|ℓn(xω,ℓn)| |
for μred-a.e. ω∈Γ.
Proof. Let ω∈Γ, and let η∈E be such that ω=red(η). We can assume that η satisfies
limn→∞1n|red(η|n)|=ℓ |
and
χ=−limn→∞1nlog|φ′red(η|n)(xη,n)|. |
Let ϵ>0, and let n∈N be sufficiently large. Then, since ω|(ℓ−ϵ)n precedes red(η|n) and |red(η|n)|<(ℓ+ϵ)n, we have
−log|φ′ω|(ℓ−ϵ)n(xω(ℓ−ϵ)n)|<nχ<−log|φ′ω|(ℓ−ϵ)n(xω(ℓ−ϵ)n)|−2ϵn⋅logλmin, |
where
λmin=min{|φ′ab(x)|:(a,b)∈Λ⋆,x∈Xb}. |
The result follows from this.
For ω∈Γ∗, we denote
Eω={υ∈¯E:ω precedes red(υ)} |
and
ˆEω={υ∈¯E:there exists n∈N s.t. red(υ|n)=ω}. |
Notice that Eω⊂ˆEω. For a∈Λ, write
qa={1−μ(Ea−1)(a−1∈Λ)1(a−1∉Λ) |
and pa=μ(ˆEa). We next prove the following crucial lemma.
Lemma 3.2. Let ω∈Γ∗ and a∈Λ be such that ωa∈Γ∗. Then we have
pa=μ(ˆEωa)μ(ˆEω). |
Proof. Fix such ω∈Γ∗ and a∈Λ. Notice that
ˆEω=∞⨆i=|ω|ˆE(i)ω, |
where
ˆE(i)ω={υ∈¯E:red(υ|i)=ω, red(υ|k)≠ω (|ω|≤k<i)}. |
Then, since
ˆEωa∩ˆE(i)ω={υ∈¯E:σi(υ)∈ˆEa}∩ˆE(i)ω, |
we have
μ(ˆEωa∩ˆE(i)ω)=pa⋅μ(ˆE(i)ω). |
Therefore,
μ(ˆEωa)=μ(∞⨆i=|ω|ˆEωa∩ˆE(i)ω)=∞∑i=|ω|pa⋅μ(ˆE(i)ω)=pa⋅μ(ˆEω). |
Similarly, we have the following:
Lemma 3.3. For a∈Λ and ω=ω0⋯ωn∈Γ∗ with ωn=a, we have
μ(Eω)=qa⋅μ(ˆEω). |
Proof. Fix such a∈Λ and ω∈Γ∗. Notice that
ˆEω=∞⨆i=|ω|ˆE(i)ω, |
where
ˆE(i)ω={υ∈¯E:red(υ|i)=ω, red(υ|k)≠ω (|ω|≤k<i)}. |
Then, since
Eω∩ˆE(i)ω={υ∈¯E:σi(υ)∉⋃b∈Λ∖{a−1}Eb}∩ˆE(i)ω, |
we have
μ(Eω∩ˆE(i)ω)=qa⋅μ(ˆE(i)ω). |
Therefore,
μ(Eω)=μ(∞⨆i=|ω|Eω∩ˆE(i)ω)=∞∑i=|ω|qa⋅μ(ˆE(i)ω)=qa⋅μ(ˆEω). |
In this section we prove Theorem 2.1. Denote
˜pa=paqa and ˜pab=pbqbqa. |
Let ˜P=(˜pab). By Lemma 3.2 and Lemma 3.3, we obtain the following.
Proposition 4.1. For all ω=ω0⋯ωn∈Γ∗, we have
μred([ω])=˜pω0˜pω0ω1⋯˜pωn−1ωn. |
The above proposition implies that μred=˜μ. Therefore, we obtain ν=˜ν. By Lemma 3.1, we have χ=ℓ˜χ. It remains to show the following lemma. Notice that hRW, ℓ and ˜h all depend only on Λ and p.
Lemma 4.1. We have
hRW=ℓ˜h. |
Proof. Let Φ′={ϕab}(a,b)∈Λ⋆ be an IFS with inverses that satisfies the separation condition. Then, by Proposition 2.2 and Proposition 2.3 we obtain hRW=ℓ˜h.
The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.
The author would like to thank the anonymous referees for the careful reading and all the helpful suggestions and remarks. Y. T. was supported by JSPS KAKENHI grant 2020L0116.
The author declares no conflict of interest.
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