# Tagged Questions

228 views

### Heisenberg group acts on the circle

Let $H$ be a Heisenberg group, i.e. $$H=\left\langle a,b,c |[a,b]=c,[a,c]=[b,c]=1\right\rangle.$$ $H$ acts on the circle by homeomorphism which preserves the orientation. If the rotation number of ...
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### Contractibility of connected holomorphic dynamics?

Let $f$ be a function, holomorphic in $\mathbb{C}$, and $K(f)$ its non-escaping set : $$K(f) = \{ z \in \mathbb{C} : f^{(k)}(z) \nrightarrow_{k \to \infty} \infty \}$$ Question : If $K(f)$ ...
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### Dynamical properties of injective continuous functions on $\mathbb{R}^d$

Let $\varphi:\mathbb{R}^d\to\mathbb{R}^d$ be an injective continuous function. Denote by $\varphi_n$ the $n$-th iterate of $\varphi$, i.e. $\varphi_n(x)=\varphi_{n-1}(\varphi(x))$ for all ...
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### Extending the hyperbolic splitting on $\Lambda$ to a neighborhood of $\Lambda$

Let $M$ be a compact Riemannian manifold and let $f:M→M$ a diffeomorphism. Let $\Lambda\subset M$ be a compact invariant subset of $M$. We say that $\Lambda$ is a hyperbolic set for $f$ when there ...
358 views

### “is topologically mixing” vs. “is topologically transitive” in the defition of chaos

This question is cross-posted from MSE, since it hasn't gotten an answer there for over 72 hours. Wikipedia gives essentially "is topologically mixing and has dense periodic periodic orbits" as the ...
203 views

### inverse problem for ergodic measures

It is a basic fact in the weak-* topology, the set of invariant measures for a dynamical system is closed, compact, and convex in the weak-* topology. Furthermore, the set of ergodic measures is equal ...
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### Is a closed set with orbit capacity zero automatically thin?

Let $G$ be a countably infinite amenable group. Let $\alpha: G\curvearrowright X$ be a continuous group action. (Mostly free and minimal, though!) Definition 1: Let $A\subset X$ be closed and ...
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### Equivariant zero dimensional extension recovering a given measure

Let $X$ be a compact metrizable space and $\alpha: \mathbb{Z}^d\curvearrowright X$ a continuous group action. Then it is well known that there exists a zero dimensional compact space $Y$, an action ...
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### When does a homeomorphism split into essentially minimal homeomorphisms?

Background Suppose $X$ is a compact metric space, and that $\varphi: X\to X$ is a homeomorphism of $X$. We say a subset $A$ of $X$ is $\varphi$-invariant if $\varphi(A) = A$. A $\varphi$-invariant ...
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### Beautiful examples of arc-like continua

A continuum is a nonempty compact, connected metric space. A continuum $X$ is called arc-like if, for every $\varepsilon>0$, there is a continuous and surjective function $f:X\to [0,1]$ such that ...
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### The identity element of a compact group is a limit point of any “polynomial sequence”

Is there an "elementary" (say ultrafilter-free) proof of the following fact: if $G$ is a compact (Hausdorff) topological group, if $g \in G$ is any element from this group, and if $P$ is a polynomial ...
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### Generalized Skorokhod spaces

Skorokhod spaces of càdlàg functions are an extremely useful setting to describe stochastic processes. I'd like to understand the Skorokhod topology from a pure topological point of view, without ...
164 views

### Fundamental group of non-Hausdorff surfaces & actions of discrete Heisenberg group

Let $G$ be a discrete group, acting on a space $X$ (by homeomorphisms). I will say that the action is properly discontinuous if for any $x, y \in X$, there are neighborhoods $U_x$ and $U_y$ such that ...
471 views

### What information can one recover from the induced map on homology?

The following question came up while constructing delay embeddings of time series data. Consider an unknown topological space $X$ and an unknown continuous function $f:X \to X$. We are given a ...
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### Does the “measure-preserving property” commute with ultralimits ?

Let $(X, \mathcal{B}, \mu, T)$ be a measure-preserving system, with $T$ invertible, where the $\sigma$-algebra $\mathcal{B}$ is a Borel algebra arising from a topology which makes $T$ continuous, and ...
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### Possible homogeneity of infinite dimensional Sierpinski carpet analogues?

Start with the Hilbert cube $H=I^\omega$, thinking of its coordinates as written in ternary expansion. Construct subsets $S_n$ by removing points from $H$ if for any $m$, at least $n$ of the ...
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### Conditions under which a given scheme converges

I'm sorry in advance for how long this question is. Suppose I have a continuous function $f:\mathbb{R}^n \rightarrow \Delta_{n-1}$, where we think of the simplex $\Delta_{n-1}$ as the set ...
Suppose X is a complete metric space, and $f:X↦X$ a continuous surjective function. Let D be a dense set. Suppose $f:D↦D$ is injective and $f^{-1}(D)=D$. Is $f$ injective ? Is there a family of ...