# Tagged Questions

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In his paper Ergodic methods in additive combinatorics, Bryna Kra said that the Kronecker factor $(Z_1, \mathcal{Z}_1, m, T)$ of $(X, \mathcal{X},\mu,T)$ is the sub-$\sigma$-algebra of $X$ spanned by ...
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### Does any measure preserving system have an invertible extension?

Let $\mathsf{X} = \left\{ X,\mathcal{B},\mu,T \right\}$ be any measure preserving system. A sub-$\sigma$-algebra $\mathcal{A}\subseteq \mathcal{B}_X$ with $T^{-1}\mathcal{A}=\mathcal{A}$ modulo $\mu$ ...
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### How do you get a measure space out of a dynamical system?

I'm reading a book on ergodic theory (by Cesar Silva), and also have read Stein and Shakarchi's third book on undergraduate analysis, where there is a section devoted to some ergodic theory. Both ...
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### Ergodic Rotation of the Torus

Consider the measure preserving dynamical system $(\mathbb{R}^2 / \mathbb{Z}^2, \mathcal{B} \otimes \mathcal{B}, \lambda \otimes \lambda, R_{(\alpha, \beta)})$. This is the torus with the borel ...
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### What is “abstract” ergodic theory?

This is just a question about the usage of the term "abstract". What kind of questions in ergodic theory is considered "abstract" and what's a "regular" question? From some seminars it seems that ...
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### What is the density of the SRB measure conditioned to unstable manifolds?

I have a question regarding the SRB measure. As Lai-Sang Young puts it, the SRB-measure is the invariant ergodic invariant measure most compatible with volume. (see [ ...
Here's a passage of my script I do not understand. Define $\Omega:=\left\{z\in\mathbb{C}: \lvert z\rvert =1\right\}$ and consider $a\in [0,1)$. Then T_a:=\Omega\to\Omega, z\mapsto ... 0answers 22 views ### Unique ergodicity a spectral invariant? Suppose f:X\to X and g:Y\to Y are two automorphism of measure spaces. Suppose there is an unitary operator V:L^2(X)\to L^2(Y) satisfying VU_fV^*=U_g, here U_f and U_g are the Koopman ... 0answers 49 views ### Is there a characterization of the shift-invariant ergodic measures? Consider probability measures \mu on the space \{0,1\}^\mathbb{N} that are shift-invariant with respect to the left-shift map. Is there a nice characterization of the ergodic shift-invariant ... 1answer 40 views ### Koopman is not surjective I want to prove that the Koopman operator U_f : L^2 (\mu) \rightarrow L^2 (\mu) such that U_f(\phi) =\phi \circ f is not surjective. Where  \mu  is a measure preserving mapping f. I was ... 0answers 54 views ### Lyapunov-exponent and rate of convergence of continued fractions I'm writing an essay on the Lyapunov-exponents of the Gauß map \begin{align} T:[0,1]&\to[0,1],\\ x&\mapsto \begin{cases}\frac 1 x \mathrm{mod}\,1 & \text{for x>0,}\\ 0 & \text{for ... 1answer 38 views ### How can I prove this equivalence concerning ergodicity? I write you, because I have a problem to show two equivalences. But before writing them down, I give you all the definitions we had as background: (I) The Quadruple (\Omega,\mathcal{A},\mu,T) is ... 1answer 22 views ### almost periodic visits of a compact set of a circle group rotation Let \mathbb{T} be the circle group and R:\mathbb{T\rightarrow T} an irrational rotation (hence a minimal system). Suppose there exists a compact set K\subset\mathbb{T} such that for every ... 0answers 39 views ### Shift and ergodic measures Let X = \{0,1\}^{\mathbb{N}} endowed with the product topology and \sigma : \left\{ \begin{array}{ccc} X & \to & X \\ (\epsilon_n) & \mapsto & (\epsilon_{n+1}) \end{array} \right.. ... 1answer 56 views ### If g is invariant under an ergodic map then it's almost everywhere constant Let (X,B(X),\mu) be a probability space, where X is compact metrizable, and B(X) are the Borel sets. Let f:X\to X be a measurable function such that: i) \forall A\in B(X) ... 1answer 56 views ### what is so great about having an invariant measure? I am a student who just started to learn basic concepts of ergodic theory. It seems like that given a dynamical system, people are very excited to find various invariant measures of the system. But ... 1answer 60 views ### Unique Ergodic Measure From <http://mathworld.wolfram.com/ErgodicMeasure.html> "If there is only one ergodic measure, then T is called uniquely ergodic. An example of a uniquely ergodic transformation is the map ... 1answer 38 views ### compactness of the set of invariant measures Suppose that we have a dynamical system on some compact space X with discrete time space and transformation given by some \phi : X \rightarrow X. My question is when is the set Prob(\phi) of all ... 1answer 108 views ### An ergodic theorem on the circle Let S^1 be a circle (i.e. a closed 1-dim. manifold) and let F be a non-vanishing smooth vector field on S^1. Denote by (t,x) \mapsto \Phi_t^x the flow generated by F. I want to show ... 1answer 144 views ### How does chaos arise in Hamiltonian systems? I have a question about how chaos arises in Hamiltonian systems. I've been taking a upper year undergrad class on dynamics and it has focused on chaos a lot. So far however we have only seen the more ... 1answer 198 views ### The Denjoy Theorem I'm currently studying Denjoy's theorem, which says the following: Theorem If f is a diffeomorphism of S^1 with a irrational number rotation  \rho and the variation of f^{'} (denoted by ... 1answer 239 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 d\mu = \frac{d\sigma}{|| ... 1answer 206 views ### Non-ergodic measure Is there an easy way to see that if \mu and \nu are T-invariant measures on the same space X, and \mu \neq \nu, then \frac{1}{2}(\mu+\nu) is NOT ergodic? I know that ergodic measures are ... 1answer 71 views ### Eigenfunction of T^n vs. eigenfunction of T when T is ergodic Let (X, \mathcal{B}, \mu, T) be an ergodic system, and suppose f \in L^2(X, \mathcal{B}, \mu) is an eigenfunction of T^n for some integer n > 1. Is it possible to write f as a linear ... 1answer 122 views ### Ergodic theory question about the support of a measure. I am trying to work through some problems from " Ergodic theory with view towards number theory" but I am finding it a tad more difficult than expected. In particular, I have the following question: ... 2answers 127 views ### Recurrent point of continuous transformation in a compact metric space Given a metric space (X,d) and a transformation T:X\rightarrow X, a point x\in X is said to be recurrent iff it belongs to the closure of its orbit \{T(x), T^2(x),...\}: more precisely, ... 0answers 79 views ### Are geodesic flows on surfaces with negative curvature Anosov? I'm just going through the original book by Anosov, where he tries to proof this result. I don't quite understand it. So let \phi_t:TM\rightarrow TM be a geodesic flow on a compact surface M of ... 1answer 114 views ### Unfolding a Billiard Trajectory The following image is from page 1019 of http://www-fourier.ujf-grenoble.fr/~lanneau/references/masur_tabachnikov_chap13.pdf From what I understand about unfolding billiards we are representing the ... 0answers 124 views ### Bernoulli shift on S^\mathbb{Z} Why is the Bernoulli shift ergodic? I know the proof for S^\mathbb{N}. The best would be if you could argue from this fact to the integer case, so that way I can see how one usually goes from ... 1answer 185 views ### Ergodic Theory (Weak Mixing) If T is weak mixing then we know that T\times T \times \ldots \times T is also weak mixing. Does anyone know if this is true for T\times T \times \ldots? 5answers 242 views ### High-School Level Introduction to Dynamical Systems In one month I'll be giving a talk to motivated high schools students on a topic of my choice from dynamical systems and/or ergodic theory. I'm having trouble coming up with a topic compelling enough ... 1answer 40 views ### Historical behavior of the Birkhoff averages Birkoff Ergodic Theorem: Let be (X,\mathcal{A},\mu) a probability space, T:X\to X a measurable tranformation preserving \mu and f\in L^{1}(\mu), then there exists a \Sigma \subset X and ... 1answer 153 views ### Measurable Partition and Ergodic Decomposition I need some background before asking the question: Let \mathcal{P} is called a measurable partition if there is a measurable set M_0\subset M with full probability measure such that, restric to ... 1answer 25 views ### Finding a minimal set for the group G of homeomorphims that comute Let G the abelian group generated by homeomorphisms f_1,\dots,f_q:M\rightarrow M in a compact metric space that comute. Show that there is X\subset M minimal in the relation of inclusion in the ... 1answer 70 views ### pointwise ergodic theorem and mean sojourn time Let G be a group and let F_i be a sequence of finite subsets of G. Suppose G acts on a probability measure space (X,\mu) in a measure preserving way, and suppose that this action is ergodic. ... 1answer 70 views ### How to check the strong ergodicity of the SL_2(\mathbb{Z})-action on the torus? Suppose \Gamma\subset SL_2(\mathbb{Z}) is a non-amenable subgroup, especially, \Gamma=SL_2(\mathbb{Z}). Consider the natural action of \Gamma on S^1\times S^1=T^2. How to check that this ... 2answers 206 views ### Von Neumann's ergodic theorem Where can I find the proof of Von Neumann's ergodic theorem? Please, give me references or write names of books where I can find it. 2answers 92 views ### Can the time mean over a dense orbit equal the space mean for arbitrary functions? Let \varphi : M \to M be a measure-preserving map of a measure space M with measure \mu, and let f \in L^1(\mu) be arbitrary. If p is the starting point of an orbit that is dense in M, ... 1answer 87 views ### Lyapunov Exponent Suppose (X,A,\mu) a probability space, where X is a compact Riemann manifold, T:X\to X a diffeomorphism and T is a measure-preserving transformation( over the borel \sigma algebra). Prove ... 1answer 467 views ### Why are hyperbolic toral automorphisms (e.g. Arnold's cat map) ergodic? Let \varphi : \mathbb{T}^2 \to \mathbb{T}^2 be a hyperbolic automorphism of the torus, induced by a linear map A : \mathbb{R}^2 \to \mathbb{R}^2 of determinant \pm 1 with no eigenvalues of ... 1answer 423 views ### Kakutani skyscraper is infinite Karl E. Petersen's book "Ergodic Theory", chapter 2, exercise 9, on page 56 Prove that for any ergodic measure preserving transformation T:X\rightarrow X on a non-atomic probability space (X, ... 2answers 283 views ### An equivalent condition for strong-mixing For a measure-preserving (finite) system (X,\mathcal{B},\mu,T), is it correct that the following are equivalent? For every A,B\in\mathcal{B} , \displaystyle\lim_{n\rightarrow\infty}\mu(A\cap ... 2answers 66 views ### Defining an f-invariant measure Suppose I have a compact oriented manifold M with an orientation preserving self-diffeomorphism f. I wish to define a volume form on M which is invariant under f. Certainly, it is necessary ... 1answer 83 views ### Related questions on the Hausdorff dimension and local dimension of a Cantor set Suppose I is an interval on a Cantor tree with m children I', each of length \varepsilon. I have that \sum_{\hat I\text{ is a child of }I}|\hat I |^s=|I|^s \Rightarrow ... 1answer 147 views ### A question about continued fractions and Gauss map For \alpha \in (0,1), write \alpha as a continued fraction like \alpha=[a_1, a_2, \ldots] (note that the implicit 0th coefficient a_0=0 has been omitted), and let \frac{p_n}{q_n} be the ... 1answer 208 views ### Ergodic theory in mathematics and physics How is the theory of ergodic measure-preserving transformations related to ergodicity in the physical sense (which I understood as, very very roughly speaking, that a physical system is called ergodic ... 1answer 115 views ### Solution space to a functional equation This question comes from my attempts at understanding an example presented by Bill Gasarch on his blog. The example is of a continuous strictly increasing function whose derivative is zero almost ... 1answer 233 views ### Ergodicity of the First Return Map I was looking for some results on Infinite Ergodic Theory and I found this proposition. Do you guys know how to prove the last item (iii)? I managed to prove (i) and (ii) but I can't do (iii). Let ... 1answer 84 views ### Ergodicity of measure induced by generic points in Birkhoff's ergodic theorem Let X=\{0,1\}^{\mathbb{N}}, T:X\to X the shift map, and \mu a T-invariant probability measure on X. A point x \in X is generic if \lim\, \frac{1}{n}\sum_{i<n} ...
Assume that all prime periods of periodic orbits of a continuous map $f:[0:1]\to [0:1]$ are uniformly bounded (i.e. there exists N such that the prime period of every periodic orbit of f is smaller ...