1
vote
0answers
38 views

Baker's transformation: continuity, orbits of irrational and rational points

I've reading the Pugh's Analysis book and I have problems with one exercise. This says: The baker's transformation: a rectangle of dough is stretched to twice its length and folded back on itself. ...
3
votes
0answers
84 views

Intersections of two exponential curves in a plane

I am struggling to show that two exponential curves in $\mathbb{R}^n$ do not overlap except finite distinct points. Could anyone help me? Let $A\in\mathbb{R}^{n\times n}$ and ...
0
votes
0answers
18 views

How to generate a Poincare section for discrete particle trajectory?

I'm a novice when it comes to generating Poincare sections, and I can't seem to get it right. I have a particle moving in a 3D periodic field, and I wish to generate a Poincare section of its ...
1
vote
0answers
26 views

The future of the orbit of a point is a closed set [duplicate]

$X$ is a metric space and $f: X \rightarrow X$ is a dynamical system. Prove: $w(x_{0})$ is closed. Here the set $w(x_{0})$ is the future of the orbit of $x_0$, defined as $$\omega(x_0) = \{y \mid ...
0
votes
1answer
96 views

Irrational Rotation

Let $\sigma$ be a homeomorphism of $S^1$. Then the following statements are equivalent; (1) O(z) is dense in S for some z in S, (2) O(z) is dense in $S^1$ for every z in $S^1$, (3) $\sigma$ is ...
1
vote
1answer
61 views

Compact $\omega$-limit set $\Rightarrow$ connected

Consider the flow $\varphi: \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n$ and $L_{\omega}(x)$ the $\omega$-limit set of a point $x \in \mathbb{R}^n$. How can I show that if $L_{\omega}(x)$ is ...
0
votes
0answers
14 views

Topological semi-conjugacy between two systems

I have a few questions regarding the clarification of the difference between semi-conjugacy and conjugacy. I am trying to figure out a explicit conjugator function to semi-conjugate (a-x^2, x) and ...
1
vote
1answer
41 views

Finite family of subtori in the torus $(S^{1})^{n}$

Working on a problem on matroids, I've already ask a question about some subtori. Here's the link to a previous problem: Topological subspace in $(S^{1})^{n}$ Anyway, here's another problem related ...
3
votes
1answer
81 views

Real analysis with a non-standard topology

I have recently undertaken a self study of topology and am using Munkres Topology 2nd edition as the primary text. My background(theoretical chemistry & physics) is almost entirely void of any ...
3
votes
1answer
69 views

The abelian group of homeomorphism

Let $G$ be a subgroup of the group of homeomorphisms on the circle, and we suppose $G$ is abelian, if every element of $G$ has a fixed point on the circle, does it imply that $G$ has a common fixed ...
1
vote
0answers
21 views

Topological entropy, spanning sets and expansiveness of simple maps on a torus

I am trying to solve the following problem. Take the torus $\mathbb{T}^{2}$ and define the map $T(x,y)=(x + \alpha$ mod 1, $x+y$ mod $1)$, where $(x,y) \in [0,1]^{2}$. By induction, we have ...
2
votes
3answers
144 views

What is the topological dimension of the Peano curve?

The Hausdorff dimension of the Peano curve is know to be two. And I assume it to be a fractal since it's on the List of fractals by Hausdorff dimension. Moreover: According to Falconer, one of the ...
3
votes
3answers
128 views

Why is the Koch curve homeomorphic to $[0,1]$?

Henning Makholm has provided a nice proof that the limiting curve is a continuous function from $[0,1]$ to the plane. I was curios if the function is homeomorphism. A quick search gave me many sources ...
2
votes
1answer
49 views

Why do we require a finite number of subsets for self-similarity?

Here is how my text defines self-similarity: We call $M \subset \mathbb R^d$ self-similar if there are $T_1, \ldots, T_m \subsetneqq M$ and similarity maps $\alpha_1, \ldots, \alpha_m$ such that ...
0
votes
0answers
16 views

Showing that the function $C(b)$ is a compact set for $|b| < 1$

I am reading "An Invitation to Dynamical Systems", and one of the challenge problems is to prove that $C(b)$ is a compact set where $C(b)$ is defined as the set of all numbers that can be expressed in ...
3
votes
1answer
36 views

Commuting functions on the closed interval have the same value somewhere

We are given two commuting continuous functions $f,g:[0,1]\to[0,1]$. How can we prove that $f(x)=g(x)$ for some $x\in[0,1]$? A trivial observation is that if one of the two functions is a ...
1
vote
1answer
99 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? ...
1
vote
2answers
67 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
29 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
53 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
100 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
129 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
42 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
63 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
176 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
160 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 ...
4
votes
1answer
241 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
86 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
102 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
136 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
69 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
70 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
183 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
32 views

Finding a minimal non-empty closed $G$-invariant set of a compact metric space

Let $G$ the abelian group generated by commuting homeomorphisms $f_1,\dots,f_q:M\rightarrow M$, where $M$ is a compact metric space. Show that there is $X\subset M$ minimal with respect to the ...
3
votes
2answers
93 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
304 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
521 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
318 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
382 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
592 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
244 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
150 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
136 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
58 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
242 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 ...