In this paper we study ergodic measures of intermediate entropy for affine transformations of nilmanifolds. We prove that if an affine transformation τ of nilmanifold has a periodic point, then for every a∈[0,htop(τ)] there exists an ergodic measure μa of τ such that hμa(τ)=a.
Citation: Wen Huang, Leiye Xu, Shengnan Xu. Ergodic measures of intermediate entropy for affine transformations of nilmanifolds[J]. Electronic Research Archive, 2021, 29(4): 2819-2827. doi: 10.3934/era.2021015
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In this paper we study ergodic measures of intermediate entropy for affine transformations of nilmanifolds. We prove that if an affine transformation τ of nilmanifold has a periodic point, then for every a∈[0,htop(τ)] there exists an ergodic measure μa of τ such that hμa(τ)=a.
Throughout this paper, by a topological dynamical system
Given a TDS
Define
E(T)={hμ(T):μ∈Me(X,T)} |
where
It is interesting to consider the case when
[0,htop(f))⊂E(f) | (1.1) |
for any
Conjecture 1.1 (Katok). Let
We need to point out that Katok's conjecture implies that any positive entropy smooth system is not uniquely ergodic, though whether or not a smooth diffeomorphism of positive topological entropy can be uniquely ergodic is still in question (see [5] for Herman's example: positive entropy minimal
In this paper, we study intermediate entropy for affine transformations of nilmanifolds. Throughout this paper, by a nilmanifold
Theorem 1.2. Let
Following Lind [11], we say that an affine transformation of a nilmanifold is quasi-hyperbolic if its associated matrix has no eigenvalue 1. As an application of Theorem 1.2, one has the following.
Theorem 1.3. Let
The paper is organized as follows. In Section 2, we introduce some notions. In Section 3, we prove Theorem 1.2 and Theorem 1.3.
In this section, we recall some notions of entropy, nilmanifold and upper semicontinuity of entropy map.
We summarize some basic concepts and useful properties related to topological entropy and measure-theoretic entropy here.
Let
Definition 2.1. Let
htop(T,U)=limn→+∞1nlogN(⋁n−1i=0T−iU), |
where
htop(T)=supUhtop(T,U), |
where supremum is taken over all finite open covers of
A subset
hd(T,K)=limϵ→0lim supn→∞logs(T)n(ϵ,K)n. |
Let
hd(T,Z)=supK⊂ZK is compacthd(T,K). |
And the Bowen's topological entropy of a TDS
Next we define measure-theoretic entropy. Let
hμ(T,ξ)=limn→+∞1nHμ(⋁n−1i=0T−iξ), |
where
hμ(T)=supξ∈PXhμ(T,ξ). |
The basic relationship between topological entropy and measure-theoretic entropy is given by the variational principle [12].
Theorem 2.2 (The variational principle). Let
htop(T)=sup{hμ(T):μ∈M(X,T)}=sup{hμ(T):μ∈Me(X,T)}. |
A factor map
supμ∈M(X,T)π(μ)=νhμ(T)=hν(S)+∫Yhd(T,π−1(y))dν(y) | (2.1) |
where
Let
The following is from [1,Theorem 19].
Theorem 2.3. Let
Remark 2.4. (1) In the above situation, Bowen shows that
hd(T,π−1(y))=htop(τ) for any y∈Y, | (2.2) |
where
(2) If
hd(T,π−1(y))=htop(τ) for any y∈G/H, | (2.3) |
where
Given a TDS
∫Me(X,T)∫Xf(x)dm(x)dρ(m)=∫Xf(x)dμ(x) for all f∈C(X). |
We write
Theorem 2.5. Let
hμ(T)=∫Me(X,T)hm(T)dρ(m). |
We say that the entropy map of
limn→∞μn=μ implies lim supn→∞hμn(T)≤hμ(T). |
We say that a TDS
limδ→0supx∈Xhd(T,Γδ(x))=0. |
Here for each
Γδ(x):={y∈X:d(Tjx,Tjy)<δ for all j≥0}. |
The result of Misiurewicz [12,Corollary 4.1] gives a sufficient condition for upper semicontinuity of the entropy map.
Theorem 2.6. Let
The result of Buzzi [3] gives a sufficient condition for asymptotic entropy expansiveness.
Theorem 2.7. Let
In this section, we prove our main results. In the first subsection, we prove that Katok's conjecture holds for affine transformations of torus. In the second subsection, we show some properties of metrics on nilmanifolds. In the last subsection, we prove Theorem 1.2 and Theorem 1.3.
We say that a topological dynamical system
Theorem 3.1. Let
Proof. We think of
τ(x)=A(x)+b for each x∈Tm. |
Let
htop(τ)=∫Me(Tm,τ)hν(τ)dρ(ν). |
By variational principle, there exists
Case 1.
π(x)=x−q for each x∈Tm. |
Then
Case 2.
H={x∈Tm:(A−id)mx=0}. |
Then
This ends the proof of Theorem 3.1.
Let
If
We fix an
τ(gΓ)=g0A(g)Γ for each g∈G |
where
Aj:Gj−1Γ/GjΓ→Gj−1Γ/GjΓ:Aj(hGjΓ)=A(h)GjΓ for each h∈Gj−1 |
and
τj:G/GjΓ→G/GjΓ:τj(hGjΓ)=g0A(h)GjΓ for each h∈G. |
It is easy to see that
For each
πj+1(gGj+1Γ)=gGjΓ for each g∈G. | (3.1) |
It is easy to see that
Lemma 3.2. For each
Proof. In Remark 2.4 (2), we let
hdj+1(τj+1,π−1j+1(y))=htop(Aj+1)=bj+1 for every y∈G/GjΓ. |
This ends the proof of Lemma 3.2.
The following result is immediately from Lemma 3.2, (2.1) and Theorem 2.7.
Lemma 3.3. For
We have the following.
Corollary 3.4.
Proof. We prove the corollary by induction on
htop(τj+1)=supμ∈M(G/Gj+1Γ,τj+1)hμ(τj+1)≤supμ∈M(G/GjΓ,τj)(hμ(τj)+∫G/GjΓhdj+1(τj+1,π−1j+1(y))dμ(y))≤htop(τj)+supμ∈M(G/GjΓ,τj)∫G/GjΓhdj+1(τj+1,π−1j+1(y))dμ(y)=j∑i=1bi+bj+1=j+1∑i=1bi, |
where we used Lemma 3.2. On the other hand, by Lemma 3.3 there exists
Remark 3.5. We remark that the topological entropy of
htop(τ)=hd(τ)=∑|λi|>1log|λi| |
where
Lemma 3.6. For
Proof. We fix
hν(τj+1)=supμ∈M(G/Gj+1Γ,τj+1)πj+1(μ)=νjhμ(τj+1)=hνj(τj)+bj+1. |
We fix such
ν=∫Me(G/Gj+1Γ,τj+1)mdρ(m). |
Then by property of ergodic decomposition, one has
ρ({m∈Me(G/Gj+1Γ,τj+1):πj+1(m)=νj})=1. |
Therefore, for
hm(τj+1)≤hν(τj+1)=hνj(τj)+bj+1. |
Hence by Theorem 2.5, one has
hνj(τj)+bj+1=hν(τj+1)=∫Me(G/Gj+1Γ,τj+1)hm(τj+1)dρ(m)≤hνj(τj)+bj+1. |
We notice that the equality holds only in the case
hνj+1(τj+1)=hνj(τj)+bj+1 and πj+1(νj+1)=νj. |
This ends the proof of Lemma 3.6.
Now we are ready to prove our main results.
Proof of Theorem 1.2. Firstly we assume that
s+1∑j=i+1bj≤a≤s+1∑j=ibj. |
Since
τi(pGi−1Γ/GiΓ)=pγGi−1Γ/GiΓ⊂p[γ,Gi−1]Gi−1γΓ/GiΓ⊂pGi−1Γ/GiΓ, |
where we used the fact
π(phGiΓ)=hGiΓ for each h∈Gi−1. |
Then for each
π∘τi(phGiΓ)=p−1g0A(p)A(h)GiΓ=γA(h)GiΓ=A(h)γ[γ,A(h)]GiΓ=A(h)GiΓ |
where we used the fact
(pGi−1Γ/GiΓ,τi) topologically conjugates to (Gi−1Γ/GiΓ,Ai). |
Notice that
hμa(τ)=hνs+1(τs+1)=hνi(τi)+s+1∑j=i+1bj=a. |
Thus
Now we assume that
This ends the proof of Theorem 1.2.
Proposition 3.7. Let
Proof. We prove the proposition by induction on
gA(p)=gA(˜p)A(p′)=˜pˉg−1ˉgp′=˜pp′=p. |
By induction, we end the proof of Proposition 3.7.
Proof of Theorem 1.3. This comes immediately from Proposition 3.7 and Theorem 1.2.
W. Huang was partially supported by NNSF of China (11731003, 12031019, 12090012). L. Xu was partially supported by NNSF of China (11801538, 11871188, 12031019) and the USTC Research Funds of the Double First-Class Initiative.
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