6
votes
6answers
474 views

Furstenberg $\times 2 \times 3$ conjecture, bibliography

Furstenberg $\times 2 \times 3$ original conjecture states that the unique continuous invariant probability measure for $2x$ mod $1$ and $3x$ mod $1$ is the Lebesgue measure. I wanted to have a ...
4
votes
1answer
96 views

Automorphism group of compact abelian group

I am looking for references on the automorphism group $\mathrm{Aut}(X)$ of a compact abelian group $X$. By automorphisms I mean topological group automorphisms. Some particular questions are as ...
3
votes
2answers
107 views

Mixing property of first return map

Let $(X,\mathcal{X},\mu)$ be a probability measure system, $T:X\to X$ be a $\mu$-preserving isomorphism on $X$. Let $A\in \mathcal{X}$ such that $\mu(\bigcup_{n\ge 0}T^nA)=1$, and $\mu_A$ be the ...
1
vote
1answer
126 views

Physical Measure Vs. SRB measures

Anybody can help me to have an idea about an example showing the difference of a Physical measur with compare to an SRB measure? By a Physical measure i mean in the sense of $\nu$ a ...
4
votes
2answers
168 views

Invariant measures on a compact metric space

I'm dealing with a continuous flow on a compact metric space $X$, and $\mu$, $\nu$ are two invariant Borel probability measures on $X$. If I know that $\mu(A)=\nu(A)$ for all the invariant Borel ...
6
votes
1answer
176 views

Renewal systems: Intrinsic ergodicity and a question related to the Adler's conjecture

Consider the alphabet $\mathcal{A} = \{0,1\}$ and consider a finite set of words $W = \{\omega_1, \ldots , \omega_n\}$ over $\mathcal{A}$. Then the renewal system $\Sigma_{W}$ generated by $W$ is ...
3
votes
1answer
147 views

Real analytic ergodic diffeomorphisms of the two sphere

Does there exists a real analytic area preserving ergodic diffeomorphism on $S^2$? (Possibly by perturbing a rotation in the real-analytic topology?)
2
votes
1answer
107 views

Does conjugacy preserve the set of synchronizing blocks?

A synchronized system is a transitive shift space $X$ which has a synchronizing block $v$, that is $v$ is an admissible block for $X$ and whenever $vw$ and $uv$ are admissible blocks in $X$, then ...
2
votes
1answer
116 views

Mixing coded systems and period of their graph presentations

A coded system [see F. Blanchard, G. Hansel, Systèmes codés, Theoretical Computer Science, Vol. 44, 1986, pp. 17-49, http://dx.doi.org/10.1016/0304-3975(86)90108-8. ...
5
votes
2answers
211 views

Lebesgue entropy zero and positive topological entropy

I am looking for examples of volume preserving $C^{\infty}$ diffeomorphisms $f$ of a surface, which have positive topological entropy ($h(f) > 0$), but that the Lebesgue measure entropy (metric ...
3
votes
1answer
109 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 ...
11
votes
2answers
286 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 ...
11
votes
0answers
331 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 ...
2
votes
1answer
100 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), ...
3
votes
1answer
157 views

A question about ergodicity

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 ...
2
votes
1answer
98 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 ...
2
votes
0answers
71 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 ...
0
votes
0answers
106 views

Express measurable entropy in terms of Fourier coefficients of the measure

Let $S^1$ be the unit circle , $\mu$ be a Borel probability measure on $S^1$ and $T:S^1\to S^1$ is a measure-preserving map (not necessarily invertible) with respect to $\mu$. The Fourier ...
5
votes
1answer
121 views

Is there a similar theorem in the partially hyperbolic case?

Theorem 5.10.3 from Introduction to dynamical systems, by Brin & Stuck: Let $f:M\rightarrow M$ be an Anosov diffeomorphism. Then the following are equivalent: $NW(f)=M$, every unstable manifold ...
6
votes
2answers
295 views

Getting unique ergodicity from minimality

It is known that minimality does not imply unique ergodicity (Furstenberg example). I ask whether the implication holds in following particular situation: Suppose $X$ is a compact space, $f:X \to ...
2
votes
0answers
89 views

Pointwise ergodic theorem for amenable semigroups

Using tempered Folner 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 similar ...
2
votes
1answer
154 views

Uniform convergence of Birkhoff averages and unique ergodicity

I am looking for a proof or a reference for the following two facts (which appear proofless in my notes from an ergodic theory course- they might be easy but i am no expert in ET): Let $T$ be a ...
1
vote
0answers
74 views

A argument related measurable partitions in dynamic system

$X$ is a compact metric space, and $T:X\rightarrow X$ be a continuous map, which is finite to one. Denoted by$ X_{0}$ the set of all points $x\in X$, such that for all sufficiently small ...
5
votes
0answers
95 views

Using topological pressure to determine a subshift of finite type

I am interested in recognising graphs (or matrices, or subshifts of finite type) using topological pressure. Suppose that we play the following game: ${\bf Step 1:}$ I write down an irreducible n x n ...
2
votes
0answers
107 views

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 ...
6
votes
3answers
277 views

Poincare recurrence theorem and convergence on compact metric spaces

I am looking for a proof (or a reference to a proof) of the following theorem: Let $X$ be a compact metric space with metric $d$, endow $X$ with the Borel $\sigma$-algebra and a probability measure ...
5
votes
1answer
170 views

Under what conditions can interval exchanges be approximated by periodic maps?

Under what conditions can an interval exchange be approximated by periodic maps? (in the weak topology for the Lebesgue measure on $[0,1]$ ). Are there non-trivial examples of periodically ...
9
votes
0answers
342 views

Poincaré recurrence and symplectic packings

Question. Is there any example of a path connected symplectic manifold $(M,\omega)$ that has infinite volume, but which cannot be packed by an infinite number of symplectic balls of a fixed radius ...
6
votes
1answer
194 views

Dynamics of $3^x$ mod 1

Consider the map $f(x)=3^x$ mod 1. Using the the iterated function system $T_{0}x=\log_{3}(x+1), T_{1}x=\log_{3}(x+2)$ we see that $f$ is dynamical conjugated to a full shift on two symbols. Moreover ...
2
votes
1answer
89 views

whether there are some books and original papers ergodic theory approach to ODE

Recently I become more and more interested in the field of ergodic theory, especially in the dimension theory and thermal formalism and its applications. People always said that most of the ideas in ...
5
votes
1answer
192 views

inverse problem for ergodic measures

It is a basic fact in the weak-* topology, the set of invariant measures for a dynamical system is closed, compact, and convex in the weak-* topology. Furthermore, the set of ergodic measures is equal ...
6
votes
1answer
277 views

What is known about dynamics on Grassmannians?

I have found myself becoming interested in dynamical systems given by homeomorphisms acting on $G(r,d)$, the space of $r$-dimensional subspaces of $\mathbb{R}^d$. I tried to do a literature search ...
4
votes
1answer
281 views

Intuition of Kolmogorov-Sinai entropy

For a measurable entropy of measurable transformation $T$ from $(X,\mathcal{B},m)$ to itself. For each finite measurable partition $\mathcal{A}=\{A_i\}_{i=1}^{m}$ of $X$, we can define ...
3
votes
1answer
191 views

Examples of transformations that are totally ergodic but not weakly mixing?

The most natural examples are irrational rotations. Are there other examples that are fundamentally different from irrational rotations? By the way, if $T$ is totally ergodic, but not weakly mixing, ...
10
votes
2answers
388 views

Birkhoff ergodic theorem and the measure of the bad points

In the Birkhoff ergodic theorem we have a PMPS $(X,B,\mu,T)$ and that for any $f\in L^1(X,\mu)$ $\frac{1}{N}\sum_{n=0}^{N-1}f(T^n x)\to \int f \, d\mu,$ in measure, in $L^1$-norm and $\mu$-a.e. My ...
2
votes
1answer
211 views

Liouville's theorem: How to get an invariant measure?

Liouville's theorem states that the `natural' 2-from is preserved under the Hamiltonian flow. Apparently this leads to an invariant measure $\mu$ as follows \begin{equation} d\mu = \frac{d\sigma}{|| ...
5
votes
0answers
169 views

Quasicompactness of transfer operators associated to IID matrix products

Let $P^1$ denote one-dimensional real projective space, and for each $A \in GL(2,\mathbb{R})$ let $\overline{A}$ denote the homeomorphism of $P^1$ induced by $A$. I am currently reading a paper which ...
6
votes
2answers
374 views

Variational Principle for the Entropy

Theorem: Let be $f$ a homeomorphism of a compact metric space $X$, then $$ h_{top}(f)=\sup_{\mu\in \mathcal{M}_{f}}~ h _\mu (f) $$ Question: The above theorem is the famous variational principle ...
3
votes
0answers
97 views

Are irrational multiples of central sets again central?

Let me begin by giving the relevant definitions. A set $A \subset \mathbb{N}$ is said to be central if and only if there exists a topological system $(X,T)$ (with $X$ a compact metric space, $T$ a ...
0
votes
1answer
203 views

Recurrence and transience of cocycle over a dynamical system

Let $X$ be a compact metric space, $T$ a homeomorphism on $X$ and $\mu$ a $T$-invariant probability measure. Let $\phi:X\to\mathbb{R}$ be a continuous function and ...
3
votes
1answer
118 views

The relations between conservative part and conservativity

I revised the question. In smooth ergodic theory, a diffeomorphism is said to be conservative (I), if it preserves the Lebesgue measure. So for some of us, conservativity is just short for ...
5
votes
2answers
208 views

Liverani's CLT (a question)

Let $(\Omega,\mathcal{F},P)$ be a probability space where $\Omega$ is a complete separable metric space, let $T:\Omega\to \Omega$ ` be an ergodic transformation, let $\hat{T}:L^{2}_{_P}(\Omega)\to ...
4
votes
1answer
260 views

Characterising ergodicity of continuous maps

Hello all. Suppose $X$ is a Polish space, $\mu$ is a Borel probability measure on $X$, and $T:X \to X$ is a continuous $\mu$-preserving map which is not ergodic. Does there necessarily exist a Borel ...
5
votes
2answers
230 views

pointwise ergodic theorem and mean sojourn time

Originally posted on Maths StackExchange, but repositing here because of getting no answer there. Not a research question really - I'm just confused by implications between various ergodic ...
6
votes
0answers
216 views

Do ergodic isometries have discrete spectrum?

Let $X$ be a metric space, $\mu$ a Borel probability measure, and $T:X\rightarrow X$ be an ergodic measure preserving isometry. Is $(X,\mu,T)$ measure theoretically isomorphic to a minimal isometry ...
1
vote
0answers
150 views

Partitions of central sets via dynamical systems

In the book "Recurrence in Ergodic Theory and Combinatorial Number Theory", 1981, Furstenberg introduced the notion of central sets. He proved in Theorem 8.8 that in each finite partition of ...
5
votes
0answers
162 views

Fibre Mixing for Dynamical Systems

Hi all, I'm interested in understanding a fairly difficult theorem of Lindenstrauss Peres and Schlag. In that paper the authors prove that certain dynamical systems related to beta expansions and ...
1
vote
2answers
123 views

Relation between entropy of one-parameter group and single elements of this group

My question is motivated by the hypothesis of the Lindenstrauss' proof of arithmetic quantum unique ergodicity, and the answer to my question is certainly known. However, I could not find it in the ...
7
votes
2answers
440 views

Examples In Ergodic Theory and Topological Dynamics

I am currently studying basic Ergodic Theory: Invariant Measures Poincaré recurrence Theorem Invariant Measure For Continuous Transformations The Ergodic Theorems and Applications Ergodic ...
3
votes
3answers
468 views

Connection between properties of Dynamical and Ergodic Systems

Hi All While studying Topological and Ergodic Dynamics, I've got quite preplexed by the different Properties a system might have (minimality, regionally recurring, transitivity, mixing, ergodic, ...