# Tagged Questions

**0**

votes

**1**answer

224 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 ...

**4**

votes

**1**answer

171 views

### 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 $K(f)$ ...

**27**

votes

**2**answers

2k views

### 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 ...

**3**

votes

**1**answer

151 views

### 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 ...

**6**

votes

**2**answers

344 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 ...

**5**

votes

**1**answer

200 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 ...

**4**

votes

**0**answers

62 views

### 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 ...

**5**

votes

**0**answers

96 views

### 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 ...

**6**

votes

**2**answers

261 views

### 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 ...

**4**

votes

**1**answer

384 views

### 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 ...

**9**

votes

**3**answers

239 views

### 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 ...

**7**

votes

**0**answers

127 views

### 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 ...

**4**

votes

**0**answers

162 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 ...

**7**

votes

**1**answer

468 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 ...

**3**

votes

**1**answer

209 views

### 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 ...

**4**

votes

**0**answers

111 views

### 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 ...

**0**

votes

**1**answer

114 views

### 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
...

**6**

votes

**6**answers

705 views

### Bijective function on a dense 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 ...

**2**

votes

**1**answer

287 views

### Equilibria Exist in Compact Convex Forward-Invariant Sets

Theorem. Consider a continuous map $f : {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n}$ and suppose that the autonomous dynamical system $\dot{x} = f(x)$ has a semiflow $\varphi : ...

**5**

votes

**2**answers

309 views

### Complexity of a fixed point

Let $\varphi:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ be a homeomorphism of
the plane with fixed point $p$, i.e. $\varphi(p)=p$, and no other periodic
points. Let $r$ be a fixed natural number. My ...

**3**

votes

**2**answers

463 views

### $C^n$ And Forcing: Reading a Recent Paper By Kunen

While reading a recent paper by Kunen arxiv.org/abs/0912.3733, which deals with PFA and the existence of certain differentiable functions, (defined on all of $\mathbb{R}$) which map certain ...

**3**

votes

**1**answer

310 views

### Chaos in uniform spaces

Let $Dom$ be a uniform space, and $\hspace{.04 in}f$ be a continuous function from $Dom$ to itself satisfying:
For all non-empty open subsets $U$ and $V$ of $Dom$, there exists a natural
number $n$ ...

**11**

votes

**3**answers

2k views

### Permute Wada Lakes keeping the coastline intact? (still open in dim >2)

Wada Lakes are three disjoint open subsets of $\mathbb R^2$ with common boundary. Originally they were constructed by hand, but they also arise naturally in the real life, that is, theory of dynamical ...

**8**

votes

**3**answers

627 views

### How much “Morse theory” can be accomplished given only a continuous transformation of a space?

If $M$ is a Riemannian manifold and $f:M\to \mathbb{R}$ a Morse-Smale function (which is just a rigorous way to say "generic smooth function"), then Morse theory essentially recovers the manifold ...