1
vote
0answers
33 views

Demonstrating that the Mandelbrot Set is connected

I know that demonstrating the Mandelbrot Set is connected requires a non-trivial proof, and that Mandelbrot himself was fooled at first. But can it be demonstrated visually that the set is connected? ...
2
votes
0answers
62 views

Is the space of $C^r$ vector fields inducing locally uniformly bounded trajectories Baire? [migrated]

Let $\mathcal{V}$ be the space $C^r$ vector fields on a non-compact (smooth) manifold $M$. Being a subspace of $C^r(M, T M)$, it inherits the natural $C^r$ topology (i.e. the strong topology) of that ...
0
votes
0answers
23 views

Ask about a definition

Is there any definition about holographic space in mathematics? I searched in network but I didn't find
1
vote
2answers
64 views

Topology on the set of infinite sequences

Help, who can give me an example to show the following definition of $\lim$ and closed set let $\Sigma$ be a set of states, let $\Sigma^\omega$ denote the set of all infinite sequences of elements in ...
0
votes
1answer
27 views

Give an example of a homeomorphism of a compact metric space that has a dense orbit but no dense semiorbit.

I am at a loss in trying to find such a function. I know it must only be bijective and continuous to be a homeomorphism since it is a compact metric space. However, every function I've tried to ...
2
votes
0answers
46 views

Topological applications of topological entropy

I just learned topological entropy during a lecture about dynamical systems, and I wonder whether there exist purely topological applications of it.
0
votes
1answer
94 views

homeomorphism of flows wrt sets

Given two flows, $\phi_t: A \to A$, and $\psi_t:B \to B$, that are topologically conjugate, and a homeomorphism, $h: A\to B$, show the following relationships to be true. In the following, $x \in A$ ...
2
votes
0answers
109 views

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

Wikipedia gives essentially "is topologically mixing and has dense periodic periodic orbits" as the definition of chaos, and this paper shows that its (the paper's) definition of chaos is equivalent ...
2
votes
0answers
41 views

Looking for articles on postcritically finite rational maps in Russian or French

I'm looking for articles on postcritically finite rational maps. I found a few articles in English, but I can't find any in Russian or French.
2
votes
0answers
59 views

What is the relation between singular point for a function and the one in a vector field?

What is the difference between sigular point for a function and the one in a vector field? Is the derivative or divergence at the singular point must be infinity? By the way, what is the relation ...
0
votes
1answer
158 views

Bounded sets of isolated points in compact metric spaces

Context and definitions: Say $(X,d)$ is a compact metric space, with $f: X \rightarrow X$ continuous. For each $n \in \mathbb{N}$, the metric $d_{n}(x,y) = \max_{0 \leq k \leq ...
2
votes
4answers
120 views

Topological Dynamics: closure of forward orbit vs. $\omega$ limit set

Let $f: X \rightarrow X$ be continuous, where $X$ is a topological space. This forms a topological dynamical system. For $x \in X$, define $\omega (x) = \cap_{n \in \mathbb{N}} \overline{\cup_{i \geq ...
3
votes
1answer
198 views

The Denjoy Theorem

I'm currently studying Denjoy's theorem, which says the following: Theorem If $f$ is a diffeomorphism of $S^1$ with a irrational number rotation $ \rho$ and the variation of $f^{'}$ (denoted by ...
2
votes
1answer
72 views

Definition of the higher dimensional mapping tori

This is proving harder to search for than I imagined. The usual definition of a mapping torus $\mathcal{M}_h$ associated to a homeomorphism $h\colon X\rightarrow X$ on a topological space $X$ is the ...
1
vote
0answers
62 views

Homeomorphism and fixed point of $F\circ G$

If $F\in\textrm{Homeo}_+(\mathbb{T}^1)$ and $G\in\textrm{Homeo}_+(\mathbb{T}^1)$ commute and have each one a fixed point then $F\circ G$ has also a fixed point. I am wondering how we can apply the ...
3
votes
0answers
52 views

temporal operators: interpreting them topologically in a dynamic topological system

There's a paper that i've been reading recently called "Dynamic Topological Logic" which can be found at: http://individual.utoronto.ca/philipkremer/onlinepapers/DTL.pdf. I have a question about ...
2
votes
1answer
100 views

Dense orbits and invertibility

Let g be the logistic map $g(x) = 4x(1-x)$ and define $\phi(x) = \sin^2(\frac{\pi}{2}x)$, for $x \in [0; 1] $. Show that $\phi$  is invertible and   $\phi \circ f = g \circ \phi$  , where $f$ ...
6
votes
2answers
126 views

Recurrent point of continuous transformation in a compact metric space

Given a metric space $(X,d)$ and a transformation $T:X\rightarrow X$, a point $x\in X$ is said to be recurrent iff it belongs to the closure of its orbit $\{T(x), T^2(x),...\}$: more precisely, ...
1
vote
1answer
66 views

If one periodic orbit is attracting then $T$ is not topologically transitive

Any ideas, hints on the following would be great. Suppose that $T\colon X \to X$ is continuous, and there exist at least two distinct periodic orbits. Show that if one of the periodic orbits is ...
2
votes
1answer
68 views

Attracting point

Let $f:X\rightarrow X$ be continuous on a Hausdorff compact space with no isolated point. Suppose there is a $x\in X$ such that: $\exists_{p\geq1} f^p(x)=x$ ($f$ composed $p$ times) There is a ...
2
votes
1answer
159 views

Non wandering Set

Let be $T:X\to X$ a topological dynamical system, some definitions: Omega limit set: $\omega(x)=\{y: \exists (n_j)~~\text{such that}~ T^{n_j}(y)\to x \}$ Recurrent set: $\mathcal{R}(T)=\{x\in ...
0
votes
1answer
25 views

Finding a minimal set for the group $G$ of homeomorphims that comute

Let $G$ the abelian group generated by homeomorphisms $f_1,\dots,f_q:M\rightarrow M$ in a compact metric space that comute. Show that there is $X\subset M$ minimal in the relation of inclusion in the ...
3
votes
2answers
86 views

Is the product of a proximal system with itself proximal?

A topological dynamical system is a pair $(X,T)$ where $X$ is a compact metric space and $T$ is a continuous map from $X$ to itself. Two points $x,y\in X$ are said to to be proximal if for any ...
3
votes
1answer
236 views

Prove that the orbit of an iterated rotation of 0 (by (A)(Pi), A irrational) around a circle centered at the origin is dense in the circle.

I think the title of the question says it all. I unfortunately did not seem to conclude anything. Some ideas I had: It is easy to show that (given $T$ is the rotation) $\{T^n(\theta)\}$ is a set of ...
4
votes
1answer
464 views

Why are hyperbolic toral automorphisms (e.g. Arnold's cat map) ergodic?

Let $\varphi : \mathbb{T}^2 \to \mathbb{T}^2$ be a hyperbolic automorphism of the torus, induced by a linear map $A : \mathbb{R}^2 \to \mathbb{R}^2$ of determinant $\pm 1$ with no eigenvalues of ...
3
votes
1answer
291 views

Prove that the baker map $B(x)$ is chaotic.

I would like to show that $B(x)= 2x$ if $\ 0\leq x\leq 1/2$ $B(x)=2x-1$ if $\ 1/2 \leq x \leq 1$ is chaotic on [0,1]. I used the symbolic dynamics; any suggestions please? I briefly recall some ...
2
votes
1answer
353 views

Use the definition of “topologically conjugate” to prove that $F(x)=4x^3-3x$ is chaotic.

Let $V$ be a set. $f: V \rightarrow V$ is said to be chaotic on $V$ if 1) $f$ has sensitive dependence on initial condition, i.e. there exists $\delta>0$ such that, for any $x\in V$ and any ...
3
votes
3answers
583 views

Beneficial to touch theoretically deeper texts earlier in core areas e.g. analysis, algebra?

Desired future direction: Dynamical System(Chaos), PDE More beneficial to read theoretically deep, modern and masterpiece texts earlier, (e.g. levels like UTX/GTM/GSM/LNM/CSAM) ? Especially in core ...
6
votes
2answers
88 views

Sharkovskii-type results in other topological spaces?

I recently came across Sharkovskii's Theorem which asserts that if $f:\mathbb{R} \to \mathbb{R}$ is continuous and has a cycle of length $m$, then $f$ has a cycle of length $n$ for any $n$ which comes ...
2
votes
1answer
231 views

Limsup of continuous functions between metric spaces

Let me start with a simple example: Let $f_n:[0,1]\to[-1,1],x\mapsto \sin 2\pi nx$. For each $x\in[0,1]$, consider the sequence $\lbrace f_n(x):n\ge1\rbrace$ and denote by $F(x)$ the set of points of ...
3
votes
2answers
78 views

Entropy of a North South Transformation.

Let $f:\mathbb{S}^2\to\mathbb{S}^2$ be a continuous north south Transformation, in other words, the point $(0,0,1)$ is a global attractor for $f$ and $(0,0,-1)$ is a global attractor for $f^{-1}$. ...
5
votes
1answer
143 views

Entropy of a Linear Toral Automorphism

I'm trying to calculate the entropy of the Linear Toral Automorphism induced by $$f(x,y,z)=(x,y+x,y+z)$$ This is an exercise in the Katok book. This map has all eigenvalues ​​equal to 1. But I do ...
1
vote
1answer
132 views

Dynamics Question

Let be $T_{\beta}:[0,1]\to [0,1]$ defined by $T_{\beta}(x)=\beta x \bmod 1$ where $\beta \in (1,2).$ Questions: $T_{\beta}$ is topologically transitive? What about the periodic points? ...
2
votes
1answer
54 views

Product of Transitive Systems

Let be $M$ a topological space, and $f:M\to M$ a danymical system, i.e, a continuous map between from $M$ to $M$. We say that a dynamical system, $f:M\to M$ is topologically transitive when, ...
2
votes
2answers
240 views

Broken glass geometry

If topology is called rubber-sheet geometry, would it be accurate to describe the "cut and shuffle" topic of "piecewise isometries" as broken glass geometry ? Isometry sounds more geometrical than ...