## 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.