Tag Archives: Furstenberg

Doubling map on unit circle

1. Let \tau:x\mapsto 2x be the doubling map on the unit torus. We also consider the uneven doubling f_a(x)=x/a for 0\le x \le a and f(x)=(x-a)/(1-a) for a \le x \le 1. It is easy to see that the Lebesgue measure m is f_a-invariant, ergodic and the metric entropy h(f_a,m)=\lambda(m)=\int \log f_a'(x) dm(x)=-a\log a-(1-a)\log (1-a). In particular, h(f_a,m)\le h(f_{0.5},m)=\log 2 =h_{\text{top}}(f_a) and h(f_a,m)\to 0 when a\to 0.

2. Following is a theorem of Einsiedler–Fish here.

Proposition. Let \tau:x\mapsto 2x be the doubling map on the unit torus, \mu be an \tau-invariant measure with zero entropy. Then for any \epsilon>0, \beta>0, there exist \delta_0>0 and a subset E\subset \mathbb{T} with \mu(E) > 0, such that for all x \in E, and all \delta<\delta_0: \mu(B(x,\delta))\ge \delta^\beta.

A trivial observation is \text{HD}(\mu)=0, which also follows from general entropy-dimension formula.

Proof. Let \beta and \epsilon be fixed. Consider the generating partition \xi=\{I_0, I_1\}, and its refinements \xi_n=\{I_\omega: \omega\in\{0,1\}^n\} (separated by k\cdot 2^{-n})….

Furstenberg introduced the following notation in 1967

Definition. A multiplicative semigroup \Sigma\subset\mathbb{N} is lacunary, if \Sigma\subset \{a^n: n\ge1\} for some integer a. Otherwise, \Sigma is non-lacunary.

Example. Both \{2^n: n\ge1\} and \{3^n: n\ge1\} are lacunary semigroups. \{2^m\cdot 3^n: m,n\ge1\} is a non-lacunary semigroup.

Theorem. Let \Sigma\subset\mathbb{N} be a non-lacunary semigroup, and enumerated increasingly by s_i > s_{i+1}\cdot. Then \frac{s_{i+1}}{s_i}\to 1.

Example. \Sigma=\{2^m\cdot 3^n: m,n\ge1\}. It is equivalent to show \{m\log 2+ n\log 3: m,n\ge1\} has smaller and smaller steps.

Theorem. Let \Sigma\subset\mathbb{N} be a non-lacunary semigroup, and A\subset \mathbb{T} be \Sigma-invariant. If 0 is not isolated in A, then A=\mathbb{T}.

Furstenberg Theorem. Let \Sigma\subset\mathbb{N} be a non-lacunary semigroup, and \alpha\in \mathbb{T}\backslash \mathbb{Q}. Then \Sigma\alpha is dense in \mathbb{T}.

In the same paper, Furstenberg also made the following conjecture: a \Sigma-invariant ergodic measure is either supported on a finite orbit, or is the Lebesgue measure.

A countable group G is said to be amenable, if it contains at least one Følner sequence. For example, any abelian countable group is amenable. Note that for amenable group action G\ni g:X\to X, there always exists invariant measures and the decomposition into ergodic measures. More importantly, the generic point can be defined by averaging along the Følner sequences, and almost every point is a generic point for an ergodic measure. In a preprint, the author had an interesting idea: to prove Furstenberg conjecture, it suffices to show that every irrational number is a generic point of the Lebesgue measure. Then any other non-atomic ergodic measures, if exist, will be starving to death since there is no generic point for them 🙂

Minimal but non-ergodic volume-preserving systems

In this post we will describe the example constructed by Furstenberg, a volume-preserving diffeomorphism f\in\mathrm{Diff}^{\omega}_m(\mathbb{T}^2) which is minimal, but not ergodic. See also Parry’s book Topics in Ergodic Theory.

Let \alpha be an irrational number and R_\alpha:\mathbb{T}\to\mathbb{T} be the irrational rotation. Let r:\mathbb{T}\to\mathbb{R} be a smooth function, which induces a skew-product f:\mathbb{T}^2\to\mathbb{T}^2, (x,y)\mapsto (x+\alpha,y+r(x)). Consider the following cohomological equation:

(*)      \phi(x+\alpha)=e^{2\pi ik\cdot r(x)}\cdot\phi(x),

(⋆)      \phi(x+\alpha)=k\cdot r(x)+\phi(x).

Remark 1. Viewed (⋆) as a real-valued equation, there is an obstruction for it to admit any solution since r_0=\int r(x)dx >0. But viewed as a \mathbb{T}-valued function, the obstruction is trivial for k if k\cdot r_0 is an integer. (also nontrivial if r_0 is an irrational..)

Proposition 1. If f is not minimal, then the equation (*) has a continuous S^1-valued solution for some k\ge1.

Proposition 2. If the equation (*) has some measurable solution for some k\neq 0, then f is not ergodic.

Remark 2. If r(x)\equiv0.5, then such f is far from minimal and (⋆) has no real-valued solution for all d\neq0. But k\cdot r_0 is an integer for even number k and the equation (*) for such k admits (trivial) constant solutions.

Theorem. By a suitable choice of \alpha and r, the above equation has a L^2-solution, but no continuous solution. In particular the corresponding system is minimal but not ergodic.

Remark 3. For circle homeomorphism f on \mathbb{T}, it has only one rotation number \rho. If \rho is irrational, then f is semiconjugate to a rotation. If $f$ is C^2, then it is transitive and is actually conjugate to a rotation. A natural question is whether Denjoy’s theory admits a reasonable generalization in higher dimensions. A homeomorphism of a higher dimensional torus does not, in general, have a unique rotation vector. Even if this is the case, the minimality of the corresponding translation does not imply the
existence of a continuous semi-conjugacy to it.

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