# Questions tagged [topological-dynamics]

Topological dynamical system, i.e. a topological space, together with a continuous transformation, a continuous flow, or more generally, a semigroup of continuous transformations of that space

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### When is a composition of homeomorphisms topologically transitive provided one of the two is?

Suppose that $S$ is a (connected) regular surface, possibly with boundary, for instance an annulus. Suppose that both $f$ and $g$ are homeomorphisms whose restriction to $\partial S$ is the identity. ...

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### State of the art on: "If a dynamical system is Li-Yorke chaotic, does there exist a Cantor scrambled set?"

The following problem is presented in the paper Recent development of chaos theory in topological dynamics - by Jian Li and Xiangdong Ye :
"If a dynamical system [$(X,f)$, $X$ metric space, $f$ ...

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### Examples of expansive homeomorphisms with the specification property that are neither symbolic nor factors of mixing SFT nor product of thereof

I am looking for nontrivial examples of expansive homeomorphisms with the specification property on compact metric spaces. Here, by a ``trivial'' example I understand a subshift with the specification ...

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### Surjectivity for distal continuous functions on a compact metric space

Where can I find a proof that a distal continuous function of a compact metric space is surjective?
PS:
The person asking the question Is there an elementary proof that distal maps are invertible? ...

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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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### Minimal actions commuting with amenable actions of $\mathbb{F}_2$

For a countable discrete group $G$ acting by homeomorphisms on a compact metrizable space $X$, we say that $G\curvearrowright X$ is (topologically) amenable if there exists a sequence of continuous ...

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### Measure concentrated on the $\omega$-limit set

Let $(X,F)$ be a dynamical system with $X$ a compact metric space and $F: X\to X$ continuous. By $\omega$-limit set of a subset $A\subset X$ I mean:
$$\omega(A):= \bigcap_{n=0}^\infty \left(\overline{\...

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### Is a “uniformly syndetic” dynamical system weak mixing?

Let $X$ be a compact metric space, and $\mu$ a probability measure on $X$ with $\text{supp} \ \mu = X$. Suppose $T: X \to X$ is continuous, measure preserving and uniformly syndetic in the sense that ...

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### Is a “uniformly minimal” dynamical system ergodic?

Let $X$ be a compact metric space, and $\mu$ a probability measure on $X$ with $\text{supp} \ \mu = X$. Suppose $T: X \to X$ is continuous, measure preserving and uniformly transitive in the sense ...

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### Roelcke precompactness and Ramsey property

A survey by Nguyen Van Thé (2014) has Conjecture 1,
which is that
"every closed oligomorphic
subgroup of $S_∞$ should have a metrizable universal minimal flow with a generic
orbit." Later, ...

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### Is there an elementary proof that distal maps are invertible?

Let $T: X \to X$ be a continuous map on a compact metric space $X$. We say $T$ is distal if $\inf_n d(T^n x, T^n y) = 0$ implies $x = y$.
Then it is true that $T$ is bijective.
Question: Is there an ...

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### Are topologically free and essentially free equivalent for minimal spaces with invariant measures?

Suppose $G$ is a discrete group acting by homeomorphisms on a compact Hausdorff space $X$, such that the action is minimal. Fix an invariant Radon measure $\nu$ on $X$. Is topologically free (the ...

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### Non-minimal system in which every point is a full entropy point

Is there a discrete topological dynamical system $(X,f)$, where $X$ is a compact metric space (with distance $d$), which is transitive but not minimal, such that $h(f)>0$ and every point is a full ...

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### Is it true that $(X,T^k)$ minimal for all $k\geq1$ implies $\mathrm{Aut}(X,T) = \mathrm{Aut}(X,T^k)$ for all $k\geq1$?

Let $(X,T)$ be a topological dynamical system ($X$ is compact metric space and $T\colon X\to X$ a homeomorphism). Recall that its automorphism group is
$$ \mathrm{Aut}(X,T) = \{g\colon X\to X : \text{$...

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### Examples of minimal almost 1-to-1 extension of torus having positive entropy?

It is well known that Toeplitz subshifts are minimal almost 1-to-1 extensions of an odometer, and that some of these subshifts have positive entropy. Thus, even if a system is an almost 1-to-1 ...

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### Is a set over which dynamics are topologically conjugate to a shift map on two symbols always repelling?

Consider the one-sided full shift map $\sigma$ and the associated shift space of infinite sequences in two letters $\{0,1\}^\mathbb{N}$ on which the shift map acts, equipped with the usual metric. ...

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### What are some foundational authors/papers in dynamical systems?

I have just begun my first dynamical systems class, and I would like to try out the advice in the top answer here. To summarize, the answer suggests that when studying a new field, one should look at ...

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### Ratner's orbit closure for a unipotent semigroup

For Ratner's orbit closure theorem, one may refer to the following Wikipedia page.
Let $\{u_t\mathrel: t\in \mathbb{R} \}\subset G$ be a unipotent one-parameter subgroup of a connected Lie group. Let $...

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### Run-away functions

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous function. We say that f has the run-away property if for every compact subset $K\subseteq \mathbb{R}$ there is some positive integer N such ...

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### Non-compact dynamical systems

In topological dynamics, most of the time, we consider the continuous action of a (semi)group $G$ on a compact Hausdorff space $X$. In this context, we can envelop the group in a compact left ...

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### Center-stable manifold theorem on manifold with boundary

I would like to see if there is a Center-stable manifold theorem on the phase space that is a manifold with boundary.
Suppose $f:M\rightarrow M$ is a diffeomorphism, according to Theorem III.7 in "...

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### Injectivity of a locally strictly expanding map on a compact space

Prove that any locally strictly expanding map on an infinite compact metric space is non-injective.

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### Existence of continuous map on real numbers with dense orbit?

Does there exist a continuous map $f:\mathbb{R}\rightarrow \mathbb{R}$ such that the forward orbit of 0 is dense in $\mathbb{R}$?

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### A reference to the fact that a topologically transitive action of a group on a compact metrizable space has a dense orbit

I need a proper reference to the following obvious fact:
An action of a group $G$ on a nonempty compact metrizable space $K$ is topologically transitive (= the orbit $GU$ of any nonempty open set $...

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### Decomposition into distal and proximal

For a topological group $G$ and a bounded real- or complex-valued function $f$ on $G$, the orbit closure of $f$ is the pointwise closure in the space of all bounded functions on $G$ of the orbit of $f$...

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### Group actions and "transfinite dynamics"

I have a question about what I shall name here "transfinite dynamics" because it involves iterating a topological dynamical system $G \curvearrowright X$ through ordinals. I do not know whether this ...

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### How to determine the family of bounded functions from an infinite Fort space to $[0,1]$?

Definition: Let $X$ be a topological space and $b\in X$. We call $X$ a Fort space (with particular point $b$), when $X$ has topology $\{A\subseteq X: b \not\in A \; \text{or} \; X\setminus A\; \text{...

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### Reference request: Dynamical systems

I’m currently reading Brin and Stuck’s Introduction to Dynamical Systems, and I think I like the field a lot so far. I haven’t finished it quite yet, but what are some other good textbooks I can read ...

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### Conditional expectation with respect to a topological factor map

Let $\pi: (X,T) \to (Y,S)$ be a factor map of minimal topological dynamical systems, and let $U \subseteq X$ be open, non-empty.
Question: Does there exist a $T$-invariant, Borel probability measure $...

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### Closed free subgroups of the automorphism group of the countable atomless boolean algebra

Let $\mathcal{B}$ be the (unique up to isomorphism) countable atomless boolean algebra, and $\mathrm{Aut}(\mathcal{B})$ its automorphism group, with pointwise convergence topology.
My question: Does $...

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### Topological full groups and minimal orbit closures

Let $X$ be the Cantor set, and let $g$ be a minimal homeomorphism of $X$. Let $h$ be a homeomorphism in the topological full group of $g$, that is, for every $x \in X$, there is a neighbourhood of $x$...

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### Topological transitivity for a self-map of $\mathbb{R}$ with finitely many discontinuities

I started working with a map $f:\mathbb{R} \to \mathbb{R}$ such that it is continuous except on a finite set. I started looking for a definition of topological transitivity and topological mixing in ...

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### Equivalent Idempotents in the Ellis Semigroup

Let $(X,T)$ be a dynamical system where $T$ is a (at least countably infinite) group acting on a compact Hausdorf space $X$, and let $E(X)$ be the Ellis semigroup of this system (if we abuse notation ...

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### When is every orbit closure uniquely ergodic?

Given a topological dynamical system $(X,T)$ (so that $T$ is a homeomorphism of the compact metric space $X$) and a point $x\in X$ we call the set ${\mathcal O}(x):=\overline{\{T^nx:n\in\mathbb Z\}}$ ...

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### Ergodicity and dense orbits

Consider a compact separable Hausdorff space $X$ endowed with a finite
Radon measure $\mu$ of full support and a continuous measure-preserving
ergodic transformation $T$. Is there a dense orbit for ...

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### Can the full shift be embedded in a flow?

Write $I=[0,1]$, and let $S$ be the shift on $X=\{ (x_n)_{n\in\mathbb Z} : x_n\in I^k \}$. Is there a flow $\phi_t$ on $X$ with $\phi_1=S$? Here I require that $\phi_t$, for fixed $t$, is at least a ...

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### Is there a minimal, topologically mixing but not positively expansive dynamical system?

Is there a compact metric space $X$ and a function $f:X\to X$ such that the dynamical system $(X, f)$ has the following three properties?
minimal
topologically mixing (a map $f$ is topologically ...

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### Is there a topologically mixing and minimal homeomorphism on the circle (or on $\mathbb S^2$)?

The irrational rotation on the circle is both a homeomorphism and minimal but is not topologically mixing. The argument-doubling transformation on the circle is topologically mixing but is neither a ...

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### Hausdorff dimension = entropy/Lyapunov exponent for the baker's map?

Let $\Sigma=\{0,1\}^{\mathbb Z}$ and let $\sigma:\Sigma\to\Sigma$ be the left shift. Then it is well known that $(\Sigma, \sigma)$ is conjugate to the baker's map $B$ of the unit square:
$$
B(x,y) = \...

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### Approximation of topological dynamical systems?

I'm trying to find references to approximations of topological dynamical systems in the following sense:
A topological dynamical system $(X, f)$ consists of a topological space (typically compact ...

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### Random suborbits of a rotation

Let $u_n = x + n\alpha \pmod 1$ with $\alpha$ irrational. We know that $(u_n)_{n \geq 0}$ is dense in $\mathbb{R}/\mathbb{Z}$ (equivalently $(u_n)_{n \geq 0}$ visits every open interval infinitely ...

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### Is there a universal $\omega$-limit set?

For the purposes of this question, a dynamical system means a compact metric space $X$ together with a continuous map $f: X \to X$.
For $x \in X$, the $\omega$-limit set of $x$, denoted $\omega(x)$, ...