Questions tagged [ergodic-theory]
Dynamical systems on measure spaces, invariant measures, ergodic averages, mixing properties.
886
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Density of “diagonal sets” in amenable groups
Let $G$ be a countable amenable group with a (left) Følner sequence $(F_n)$. Let $\Gamma$ be a subset of $G$ that has density $1$ with respect to $(F_n)$, in the sense that
$$
\lim_{n \to \infty} \...
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Are arbitrary collections of ergodic measures "strongly mututally singular"?
I'm quite embarrassed not to know the answer to this question, but I think someone else will.
Suppose that $(X, T)$ is a topological dynamical system, and $\mathcal{E}$ is the collection of ergodic $T$...
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Furstenberg-Zimmer theorem: non-invertible systems
Questions
Is there a version of the Furstenberg-Zimmer Theorem for
non-invertible measure preserving systems?
Where can I find it?
What is the precise statement?
Background
In many works that ...
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2
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Natural extension of the Gauss map
Let $G:(0,1)\to(0,1)$ be the Gauss map, i.e., $G(x)=\left\{\frac1{x}\right\}$, which is known to act as the shift on the space of continued fraction expansions.
Question. Is there an explicit ...
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Entropy of $f^{m(x)+n}$ of full shift
Let $(X,\mu,f)$ be a two-sided full shift system. Assume that there is $t \in \mathbb{N}$ such that for every $n \in \mathbb{N}$ and $x \in X$, we can define $T(x)=f^{n+m(x)}(x)$, where $m(x) \leq t; $...
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Asymptotic behavior of a dynamical system of density functions
On September 24, 2022, I asked the question below on Mathematics Stack Exchange, linked here:
Link to question on Mathematics Stack Exchange.
I received two up-votes, but no comments or answer. I ...
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Direct proof that $g_t=\text{diag}(e^{t/m}I_m,e^{-t/n}I_n)$'s action on $\operatorname{SL}(d,\mathbb R)/{\operatorname{SL}(d,\mathbb Z)}$ is ergodic
I wonder if there are any direct proof that $g_t=\operatorname{diag}(e^{t/m}I_m,e^{-t/n}I_n)$'s action on $\operatorname{SL}(d,\mathbb R)/{\operatorname{SL}(d,\mathbb Z)}$ is ergodic (or even stronger,...
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0
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1
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WLLN for bootstrap means of stationary ergodic processes?
Setup:$\quad$
Suppose that $(X_n)$ is a stationary ergodic process with $E|X_1|<\infty$.
Given $X^{(n)}=(X_1, \dots, X_n)$, select a standard Efron bootstrap subsample $(X_{n,1}^*, \dots, X_{n,m(n)}...
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Non-recurrent points of $F(a,b)=(b,ba)$ in a compact metric group $G$
Consider a compact metric group $G$ [A compact topological group $G$ where the topology is generated by an invariant metric]. I am particularly interested in the case where $G$ is the $n$-dimensional ...
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7
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2
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Unique equilibrium states for systems without specification
Let $X$ be a compact metric space and let $f\colon X\to X$ be a continuous expansive map. Let $\mathcal{V}$ denote the space of Hölder continuous potential functions $\phi\colon X\to \mathbb{R}$, ...
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Ergodic diffeomorphisms of the circle
From the paper
Halmos, Paul R., In general a measure preserving transformation is mixing, Ann. Math. (2) 45, 786-792 (1944). ZBL0063.01889.
the following result is known: Let $(E,\Sigma, \mu)$ be a ...
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1
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Ergodicity question
Consider a dynamical system given by the system of ODE.
$$\frac{d x_i}{d t} = F_i(\mathbf{x}).$$
It seems to be a well-known fact that this system is ergodic if and only if the kernel of the Koopman ...
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2
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0
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Weakly mixing diffeomorphism
From
Halmos, Paul R., In general a measure preserving transformation is mixing, Ann. Math. (2) 45, 786-792 (1944). ZBL0063.01889.
the following result is known: Let $(E,\Sigma, \mu)$ be a measure ...
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The Logistic map have subexponential decay of correlation?
I was looking for information about the correlation decay of the logistic map, more precisely if there is any parameter for which its decay is subexponential, in which case I would like to know if it ...
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Are all quasi-regular points on Polish spaces generic points?
Let $X$ be a Polish space and $T\colon X\to X$ be a continuous map. We say that a point $x\in X$ is quasi-regular if for every bounded continous function $\varphi\colon X\to\mathbb{R}$ the sequence $...
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Using gradient descent in probability case
Suppose we have i.i.d. samples $x_i\sim N(0,\Sigma)$ and $y_i\sim x_i^T\omega^*+\xi_i,\xi_i\sim N(0,1)$ where $\omega^*$ is the fixed point of:
$$\omega_{i+1} = \omega_i − \eta\nabla_\omega f(\omega_i,...
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Ergodic decomposition of the action of a subgroup
Let $G$ be a countable abelian group and let $H \le G$ be a subgroup. Let $G \curvearrowright (X,\mu)$ be an ergodic measure preserving action on some probability space $(X,\mu)$. Now we know that the ...
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Question about an early result on the mixing of geodesic flows
Let $T_t$ be the geodesic flow on a surface $S$ of constant negative curvature, and let $M(f,t) := \langle \bar f \cdot (f \circ T_t) \rangle$, where $\langle f \rangle := \int_S f(x) d\mu(x)$ and ...
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2
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Can a diffusion have negative minimum or achieve large value at a given time?
Let $\sigma:\mathbb R_+\times\mathbb R\to [1,2]$ be measurable. Consider the SDE $dX_t = \sigma(t,X_t)dW_t$, where $X_0>0$ is independent of Brownian motion $(W_t)_{t\ge 0}$. For every $T>0$ and ...
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the definition of the topological pressure for matrices
Let $:\Sigma \to GL(d, \mathbb{R})$ be a continuous matrix cocycle over a topologically mixing subshift of finite type $(\Sigma, T)$. We denote by $\Sigma_n$ the set of addmisible words with the ...
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What dynamical properties should we expect from systems satisfying statistical ones?
Some results on probability theory can be generalized to more abstract ones in ergodic theory, for example:
the strong law of large numbers can be seen as a particular case of Birkhoff's ergodic ...
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Counterexamples to the Ahlfors measure conjecture in higher dimensions
Let $\Gamma<SO(3,1)$ be a finitely generated, discrete group of isometries of $\mathbb H^3$. By work of Agol, Calegari, Canary, and Gabai, the limit set of $\Gamma$ is either the entire sphere $S^2\...
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Complex Hölder space
I already posted this question on math.stackexchange, but got no response and was suggested to post it here.
I came across a space in an ergodic theory paper, which I am calling here a (complex) ...
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Lower bound for nonconventional ergodic averages in finite fields
Let $p$ be a sufficiently large prime number and $f\colon\mathbb{F}_{p}\to\mathbb{R}_{\geq 0}$ be a function bounded by 1 such that the average of $f$ over the finite field $\mathbb{F}_{p}$ is at ...
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Union of admissible words are subshift of finite type
Assume that $Q=(q_{ij})$ is a $k\times k$ with $q_{ij}\in \{0, 1\}.$ The two side subshift of finite type associated to the matrix $Q$ is a left shift map $T:\Sigma_{Q}\rightarrow \Sigma_{Q}$, where
...
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1
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Proof that Sturmian shift is uniquely ergodic using irrational rotation
I am finding proofs of unique ergodicity of Sturmian shifts however I want to know if there is a proof that link that to the unique ergodicity of irrational rotations through conjugacy for example or ...
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1
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Cohomology for extension problems in symbolic/topological dynamics?
Context: I know essentially nothing about cohomology of any kind, but I have a problem involving classifying obstructions to extensions of certain maps or covers, and I have heard that cohomology is ...
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In general is $\frac{d\,\mu_1}{d\,\mu_2}\circ T = \frac{d\,T\mu_1}{d\,T\mu_2}$?
Given an ergodic and non-singular dynamic system (definition provided here) $(X, \mathcal{B}, \mu_1, T)$ where $(X, \mathcal{B}, \mu_1)$ is a measure space and $T$ is a fixed transformation, we then ...
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Choosing the derivative of a flow
I am looking for something like the Franks' Lemma for flows. The celebrated Franks' Lemma states that: Let $f:M \rightarrow M$ be a $C^1$ diffeomorphism and $S=\{p_1,...,p_k\}$ be a finite set of ...
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A characterization of Shannon entropy in finite sets?
I am trying to solve a complicated probability problem related to Shannon Entropy.
Let $(E,p)$ be a finite set with a probability measure $p$ on $E$. $E^n$ is given the probability measure $p^n(x_1, .....
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Measure preserving system with only trivial eigenfunctions
I want to show that if $(X,\mathcal{X}, \mu, S)$ is a measure preserving system, then $S$ has no non-trivial eigenfunctions if and only if the spectral measures corresponding to all
non-constant ...
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For measure-preserving systems, is countable generatability of the invariant $\sigma$-algebra equivalent to almost all points being periodic?
Let $X$ be a second countable Hausdorff topological space, let $T \colon X \to X$ be a Borel-measurable map, define the $\sigma$-algebra $\mathcal{I}=\{A \in \mathcal{B}(X) : T^{-1}(A)=A\}$, and for ...
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Construction of minimal zero entropy measure-theoretically strong mixing subshift?
Does anyone know of a construction of a subshift (over $\mathbb{Z}$) which is
(1) minimal
(2) zero (topological) entropy
(3) measure-theoretically strong mixing (for some measure)?
I am in particular ...
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2
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Characterization of amenable actions
Let $(X,\mu)$ be a $G$-space, i.e. a measure space with a Borel quasi-invariant $G$-action. Say that $X$ is amenable (equivalently, that the action is amenable) if there is a $G$-fixed point in every ...
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Friedland metric entropy
I was asking if it is possible to extend the definition of topological Friedland entropy for $\mathbb{Z}^d$ continuos actions to measure preserving actions.
The topologica Friedland entropy is ...
CommunityBot
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Relation between the distance projective maps and their angles
Let $f:N \to \mathbb{R}^2$
be a differentiable map of smooth manifolds. Let $\mathbb{R}^2$ be decomposed as a direct sum of line bundles, i.e. $\mathbb{R}^2=E(x) \oplus F(x)$, where $F(x)$ and $E(x)$ ...
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Balls in minimal systems II
If $(X,T)$ is a minimal system uniquely ergodic with $\mu$. Is there $p\in X$ such that $\mu(\partial B(p,t))=0$ for all $t>0$ for some metric $d$ (with the same topology)?
This question is ...
CommunityBot
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Equivalent definitions of topological weak mixing
A dynamical system $f:X\to X$ is said to be topologically transitive if for any two nonempty open sets $U,V$ there exists $n \in \mathbb{Z}$ such that $f^{\circ n}(U) \cap V \neq \emptyset$. The ...
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A metric for Grassmannians
I'm reading an article by Ricardo Mañé, "The Hausdorff dimension of horseshoes of diffeomorphisms of surfaces" (https://doi.org/10.1007/BF02585431). I'm having a technical problem. Sorry for ...
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Extension of Khintchine's recurrence in a simple case
Suppose an ergodic system $(X,\mathcal{B},\mu,T)$ has a Kronecker factor that is isomorphic to an ergodic rotation, say on the Torus.
How can one prove that the large intersection property holds for $...
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Refined equidistribution for the periodic trajectories of Anosov flows?
Duke, and Linnik before him under a restrictive condition, proved that the set of closed geodesics of a given length $L$ is equidistributed on the modular surface as $L \to \infty$. This is a theorem ...
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Connection between properties of dynamical and ergodic systems
While studying topological and ergodic dynamics, I've got quite perplexed by the different properties a system might have (minimality, regionally recurring, transitivity, mixing, ergodic, uniquely ...
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Ergodicity of the action of $\operatorname{SL}(n,\mathbb R)$ on $\operatorname{SL}(n,\mathbb R)/\operatorname{SL}(n,\mathbb Z)$
$\DeclareMathOperator\SL{SL}$Let $G\mathrel{:=}\SL(n,\mathbb R)$ and $\Gamma\mathrel{:=}\SL(n,\mathbb Z)$. Consider the action of $G$ on $(G/\Gamma,\mu)$ by left translation, where $\mu$ is the Borel ...
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Ergodic transformations, their Poisson suspensions and their Krieger types
Let $T$ be an ergodic nonsingular tranformation of a Lebesgue space. Suppose that the Poisson transformation $T^*$ of $T$ is well-defined and ergodic. Denote by $\alpha$ the Krieger type of $T$ and by ...
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Maximal ergodic inequality
A map $f: X \to X$ preserves an ergodic probability $\mu$, i.e., $\mu \circ f^{-1}=\mu$ and for any $\phi: X \to \mathbb{R}$ with $\int \phi d\mu=0$,
$$\frac{1}{n} \sum_{i \le n} \phi \circ f^i \to 0 \...
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Measure preserving transformation that makes two partitions independent
I am looking for a reference for the following result. I think it is well known but I haven't seen it written down anywhere.
Let $(X, \mathcal{B}, \mu)$ be a standard measure space and let $\mathcal{...
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Mañé's example of an attractor with no natural measure
I'm reading Milnor's notes on dynamical systems and in Lecture 3 he gives an example of an attractor with no natural measure, which he attributes to Mañé. I can find no other reference in which this ...
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SRB measure and Gibbs u-state
I have been reading the famous paper of Alves, Bonatti, and Viana where they proved that there is an SRB measure for partially hyperbolic systems. Since I am new to this field, I have some basic ...
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Almost periodic functions in weak mixing extension
In Theorem 3.4.6 of the 'Lecture notes on ergodic theory' by Jesse Peterson, it is shown that in a weak mixing extension, every almost periodic function is trivial. I have a doubt in the proof of this ...
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3-periodic point implies positive topological entropy
When I learn some basic ergodic theory, I encounter an interesting exercise. As we all know, 3-periodic point often means chaos. Therefore, when a continuous map has a 3-periodic point, it may have ...
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