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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References on semigroup actions

I posted this question on Math Stack Exchange about 10 days ago, but received no answer (https://math.stackexchange.com/q/4843881/1223994). I would like to ask for references on semigroup actions on ...
Marco Farotti's user avatar
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1 answer
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Chain components and posets

Let $(X,f)$ be a topological dynamical system ($f$ continuous, $X$ compact, metric with distance $d$). Let $C\subseteq X^2$ indicate the chain recurrence relation: $$xCy\iff \forall \epsilon>0\ \...
Alessandro Della Corte's user avatar
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Existence of a minimal, weakly mixing and Lipschitz selfmap?

I am looking for an example of a dynamical system $(M,f)$ such that: $M$ is a metric space; $f:M \to M$ is Lipschitz; $f$ is weakly mixing (that is $f \times f$ is topologically transitive) $f$ is ...
Baguette's user avatar
5 votes
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Weak mixing and entering time

Let $X$ be a compact metric space and $f$ a continuous map from $X$ to $X$. Is it true, that if $f$ is weakly mixing, then the entering time $$N(U,V) = \{n \in \mathbb{N}\mid f^n(U) \cap V \neq \...
Matěj Moravík's user avatar
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Has this metric been considered anywhere?

I posted this on math stack exchange some 10 days ago, but received no answers (https://math.stackexchange.com/q/4773194/1223994). Let $X$ be a compact metric space and denote by $d$ the metric on $X$....
Marco Farotti's user avatar
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Graph-theoretic quasi-crystals?

I have recently been interested in the following purely graph-theoretic notion that weakens the assumption of transitivity in a similar way to how quasi-crystals have "(possibly) aperiodic long-...
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Dynamical systems with disjoint $\omega$-limits of single points

For $X$ compact metric spaces and $f:X\to X$ continuous, is there a nice characterization of the systems $(X,f)$ for which, for every pair of points $x,y\in X$ with disjoint orbits, we have $\omega(x)\...
Alessandro Della Corte's user avatar
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First visit of intervals for an irrational rotation

I suppose that what I look for is known, but I can't find it. Let $\left\lbrace I_n=[a_n,b_n)\right\rbrace$ and $\left\lbrace J_n=[b_n,c]\right\rbrace$ ($n\in\mathbb{N}$) be two countable families of ...
Alessandro Della Corte's user avatar
3 votes
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General references on dynamics of continuous, piecewise linear interval maps

I want to find a general reference on topological and measurable dynamics of continuous piecewise linear interval maps. I am particularly interested in cases with only three pieces. Even ...
Marco Farotti's user avatar
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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. ...
Alessandro Della Corte's user avatar
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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, $f$ ...
Marco Farotti's user avatar
6 votes
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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 ...
Dominik Kwietniak's user avatar
1 vote
1 answer
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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 ...
Ronnie Pavlov's user avatar
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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 ...
Shirly Geffen's user avatar
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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{\...
Alessandro Della Corte's user avatar
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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 ...
Nate River's user avatar
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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 ...
Nate River's user avatar
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3 votes
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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, ...
Ochotona's user avatar
44 votes
3 answers
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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 ...
Nate River's user avatar
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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 ...
Dan Ursu's user avatar
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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 ...
Alessandro Della Corte's user avatar
4 votes
1 answer
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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{$...
RB1995's user avatar
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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 ...
RB1995's user avatar
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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. ...
aghostinthefigures's user avatar
15 votes
3 answers
737 views

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 ...
Ovi's user avatar
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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 $...
Kwok Kin Wong's user avatar
3 votes
1 answer
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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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7 votes
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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 "...
Xiiao's user avatar
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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.
yogamat's user avatar
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19 votes
2 answers
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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 $\mathbb{R}$?
yogamat's user avatar
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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 $...
Taras Banakh's user avatar
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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 $f$...
user95282's user avatar
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15 votes
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Group actions and "transfinite dynamics"

$\DeclareMathOperator\Sym{Sym}$I have a question about what I shall name here "transfinite dynamics" because it involves iterating a topological dynamical system $G \curvearrowright X$ ...
Burak's user avatar
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2 votes
1 answer
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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{...
Zeno cosini's user avatar
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3 answers
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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 ...
James Baxter's user avatar
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1 vote
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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 $...
Dan Glasscock's user avatar
4 votes
3 answers
251 views

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 $...
Iian Smythe's user avatar
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1 vote
1 answer
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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 $x$...
Colin Reid's user avatar
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1 vote
0 answers
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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 ...
Rafael Alcaraz Barrera's user avatar
4 votes
0 answers
141 views

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 ...
Simon_Peterson's user avatar
10 votes
2 answers
427 views

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 ${\mathcal O}(x):=\overline{\{T^nx:n\in\mathbb Z\}}$ ...
Joel Moreira's user avatar
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6 votes
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605 views

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 ...
coudy's user avatar
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9 votes
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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 ...
Christian Remling's user avatar
2 votes
1 answer
538 views

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 ...
user's user avatar
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6 votes
2 answers
619 views

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 ...
user's user avatar
  • 83
3 votes
0 answers
266 views

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) = \...
Nikita Sidorov's user avatar
7 votes
1 answer
371 views

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 ...
Giraffro's user avatar
  • 141
7 votes
2 answers
319 views

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 ...
Stéphane Laurent's user avatar