# Tagged Questions

Dynamical systems on measure spaces, invariant measures, ergodic averages, mixing properties.

158 views

### Automorphisms of strictly ergodic shift spaces

Let $X$ be a strictly ergodic shift space, and $\omega_1$, $\omega_2$ be two different points in $X$. Is there an automorphism $\Psi$ of $X$ such that $\Psi(\omega_1)=\omega_2$? By an automorphism I ...
597 views

### Are rounded rectangle billiard dynamics ergodic?

Bunimovich proved that the billiard-ball dynamics in the Bunimovich stadium is ergodic.            (Image from this link.) Q. Is it known that the ...
471 views

### Blocking light with mirrored convex objects

There is a long-unsolved problem posed by Janos Pach, sometimes known as the enchanted forest problem, which asks if it is possible to block a point light source in the plane from reaching infinity by ...
210 views

### Is irreducibility sufficient for uniqueness of invariant distribution for a Feller Semigroup?

Let $(T_t)$ be a strongly continuous semigroup of positive operators on $C(K)$, where $K$ is a compact space. Assume also that $T_t1 =1$ for every $t\geq 0$. (This is also called a Feller Semigroup.) ...
123 views

Let $T$ be an arbitrary Lebesgue measure-preserving automorphism of the unit interval $I$. Let $R_{\alpha}$ denote rotation by $\alpha$, i.e. $R_{\alpha}(x)=x+\alpha \pmod{1}$ for $x \in I$ and $\... 0answers 63 views ### absolute continuity of a measure given absolute continuity of conditionals Situation is the following. We have the two-dimensional torus$X$and have partition$\xi$into vertical circles$\{x\} \times S^1$. We are given a measure$\mu$on$X$such that the projection ... 1answer 140 views ### the union of local stable manifolds along local unstable manifolds Let$f:M\rightarrow M$be a$C^2$hyperbolic diffeomorphism on compact connected riemannian manifold$M$. then there are local stable and unstable manifolds at each point denoted by$W^s_\delta(x), W^...
192 views

Let $X$ be a compact metric space, $T:X\rightarrow X$ a homeomorphism and $\mu$ be a $T$-invariant probability measure on $X$ such that the set of points with dense orbit in $supp(\mu)$ has full ...
159 views

### Decay of Correlation, references for a non-standard way

In ergodic theory is common to use the decay of correlation property to deduce properties analogues to those of i.i.d. random variables. Call $X\doteq [0,1].$ Examples of decay of correlation ...
101 views

### uniquely ergodic hyperbolic invariant set

The question is to classify uniformly hyperbolic invariant sets supporting uniquely ergodic invariant measure. The only examples that I expect are: fixed points, periodic orbits and Cantori(Denjoy ...
182 views

209 views

### Pointwise ergodic theorem for amenable semigroups

Using tempered Følner sequences one may show a pointwise ergodic theorem for amenable groups. (see http://www.aimsciences.org/journals/pdfsnews.jsp?paperID=2413&mode=full) Is there a ...
186 views

258 views

161 views

### Arithmetic Fuchsian lattices that are not finite index subgroups of Eichler orders?

Lindenstrauss' proof of AQUE (arithmetic quantum unique ergodicity) assumes that the Fuchsian lattice is an Eichler order or, if I understand it correctly, a finite index subgroup of an Eichler order. ...
### Reference for and Properties of the $alpha$-entropy
Let $T : X \to X$ be a continuous map on, say, a compact metric space $X$. Let $\mu$ be an invariant borel measure. Under suitable conditions, a result of Brin and Katok states that $\mu$-almost ...