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
20
questions with no upvoted or accepted answers
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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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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 ...
- 1,216
6
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135
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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-...
- 685
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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 ...
- 1,588
6
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153
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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 ...
- 2,413
6
votes
0
answers
609
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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 ...
- 18.5k
4
votes
0
answers
141
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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 ...
- 297
3
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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 $X$....
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3
votes
0
answers
144
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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 ...
- 309
3
votes
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267
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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) = \...
- 2,085
2
votes
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answers
119
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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. ...
- 4,263
2
votes
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88
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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 ...
- 265
2
votes
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answers
74
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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 "...
- 21
1
vote
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159
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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. ...
1
vote
0
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73
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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$...
- 997
1
vote
0
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84
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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 $...
- 859
1
vote
0
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82
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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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33
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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 ...
- 1
0
votes
0
answers
213
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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{\...
- 4,263
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0
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88
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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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