Questions tagged [topological-dynamics]
Topological dynamics is a subfield of the area of dynamical systems. The main focus is properties of dynamical systems that can be formulated using topological objects.
95 questions
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Existence Of An Orbit of the Tent Map whose Closure is the Cantor Set [closed]
How to prove that
$$
T(x) = \frac{3}{2} - 3|x-\frac{1}{2}|,
$$
has at least one point with dense orbit with respect to its invariant middle third Cantor set?
Are there any basic solutions?
Thank you ...
- 420
2
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89
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Closed set of periodic points implies boundedness of primitive periods
I am trying to prove the following assertion, but I don't know how to proceed.
If $(X, f)$ is a discrete dynamical system (i. e. $f: X \rightarrow X$ is a continous map on the compact metric space $X$...
- 27
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Any periodic point has period q.
Here is the question I am thinking about:
Show that if $f: S^1 \to S^1$ is a homeomorphism with rational rotation number $\frac{p}{q}$ where $p$ and $q$ are coprime positive integers, then any ...
- 3,137
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Equidistribution of orbits.
Here is a question I am thinking about:
Let $X$ be a compact metric space. Show that if $(X,T,\mu)$ is an ergodic topological probability measure preserving dynamical system and $x \in X$ is generic, ...
- 2,197
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Generic points in ergodic dynamical systems.
Suppose that $(X, T, \mu)$ is an ergodic topological probability measure preserving dynamical system on a compact metric space $X.$ I want to show that almost every point in $X$ is $\mu$ generic.
...
- 2,197
3
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1
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Ergodic T invariant measures that can not be Lebesgue measures.
Let $T: S^1 \to S^1$ be the rotation of the circle by $90$ degrees. I want to find an ergodic T-invariant measure.
My question is:
I was looking here Other invariant measures than Lebesgue measure? ...
- 2,197
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Proving that if it is Dirac measure, then it is invariant.
I was reading this question here:
Show that the only $f$- invariant probability measure is the delta measure at $0.$
And from my understanding, the meaning of "prove that the only invariant ...
- 105
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Corollary of Banach's fixed point theorem
I tried to prove this corollary of BFPT. Here's the statement :
"Let $\left( E, d \right)$ be a complete metric space and $f : E \longrightarrow E$ a mapping such that $f^p$ is a contraction for ...
- 420
4
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1
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82
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Other invariant measures than Lebesgue measure?
Consider a rational rotation of the circle. What are other invariant measures different than the Lebesgue measure?
Any hints will be greatly appreciated!
- 105
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1
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What are the $f$- invariant measures?
Here is a part of the question that I am thinking about:
Let $X$ be the unit circle in $\mathbb R^2.$ Let $A$ be a $2\times 2$ matrix with real entries, determinant 1, and which does not have finite ...
- 3,137
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Find all $\tilde{f}$-invariant measures on $S^1.$
Here is the question I am trying to solve:
let $X = \mathbb R$ and let $f: X \to X$ defined by $f(x) = x + 1.$
1- Use Poincaré recurrence to show that there are no finite $f$-invariant probability ...
- 3,137
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1
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71
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Show that the only $f$- invariant probability measure is the delta measure at $0.$
Here is the question I am trying to think about:
Let $X$ be the open unit disk in $\mathbb C$ and let $f: X \to X$ be given by $f(z) = z^2.$ Use Poincaré recurrence to show that the only $f$- ...
- 3,137
1
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1
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Proof of Lemma 4.1.3 in 'Translation Surfaces' by Masur and Athreya
I am having a hard time understanding the proof of Lemma 4.1.3. They make the assertion that $q' \in \alpha^+$, but this statement seems questionable. Here is my thinking.
We note that the flow $\phi$ ...
- 133
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0
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48
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Topological entropy equal zero in countable dynamics
Topological entropy
Let $X$ a compact metric space with metric $d$. Let $f:X \to X$ a continuous function. For every $n \in \mathbb{N}$ define
$$ d_n(x,y) = \max_{0\leq k \leq n-1} d(f^k(x), f^k(y))$$
...
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1
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When $f \times g$ is such that every point has dense orbit?
For a compact metric space $X$ and functions $f,g: X \rightarrow X$, $f \times g: X^2 \rightarrow X^2$ is defined by $(f \times g) (x,y) = (f(x),g(y))$. We also write $(f\times g)^n(x,y) = (f^n(x),g^n(...
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More on rationally independent subsets of $\mathbb{R}$.
Suppose that $\lambda_{1}, \lambda_{2}, \lambda_{3}\in\mathbb{C}\setminus\{0\}$ and that $\frac{\lambda_2}{\lambda_1}, \frac{\lambda_3}{\lambda_1}\in\mathbb{R}^{+}\setminus\mathbb{Q}$ such that the ...
- 675
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0
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Are these mathematical symbols $\omega$-limit sets? [closed]
Given a system defined by a vector field in $\mathbb{R}^2$ with isolated equilibrium solutions, which of the following mathematical symbols could be $\omega$-limit sets for some point?
$$\infty, \circ,...
3
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Iterative application of continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $\lim_{n \to \infty}f^{n}(x)$ exists $\forall x$.
The question
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that $\lim_{n \to \infty}f^{n}(x)$ exists $\forall x$. Define $S = \{\lim_{n \to \infty}f^{n}(x): x \in \mathbb{R}...
- 660
1
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1
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Non-wandering set of diffeomorphism with a unique fixed point
Let $M$ be a closed manifold and $f\in\mathrm{Diff}^\infty(M)$ be a diffeomorphism of $M$. Suppose that $f$ has a fixed point $p_0$, and that for every $p\in M$, we have $f^n(p)\rightarrow p_0$.
Can ...
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Bowen metrics induce the same topology
I'm dealing with the question proving the Bowen metrics induce the same topology. More specifically, given $(X,T)$ be a topological dynamical systems equipped with a metric $d$ ($X$ is compact, $T$ is ...
- 794
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1
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132
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Definition of orbit equivalence
I have a doubt regarding the definition of orbit equivalence as given by Fisher & Hasselblatt in their book Hyperbolic Flows.
We say that two flows $\phi, \psi$ on $X, Y$ respectively are orbit ...
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0
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A question on showing $f$ is topological transitive.
Consider $\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$ and the following unimodular matrix
\begin{equation}
A=\begin{bmatrix}
2& 1\\
1& 1
\end{bmatrix}.
\end{equation}
We know $F\colon\mathbb{R}\...
- 123
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104
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Showing that the horseshoe set is locally minimal
I'm trying to prove the Smale's horseshoe set is locally minimal.
More specifically, let $H$ be the horseshoe set described in Section 1.8 in the book "Introduction to Dynamical Systems" by ...
- 794
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0
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42
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References on semigroup actions
I would like to ask for references on semigroup actions on metric spaces from a topological point of view. Thanks
- 161
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24
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How to prove factors of distal systems are distal?
Let $(X,\Phi,d_X)$ and $(Y,\Pi,d_Y)$ be topological systems. Suppose $(X,\Phi,d_X)$ and $(Y,\Pi,d_Y)$ are distal, that is for any $x_1,x_2\in X$ with $x_1\not=x_2$, one has $$\inf\limits_{t\in\mathbb{...
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Conjugacy of expansive flows
I'm reading "Expansive one-parameter flows" by Bowen-Walters.
Let $(X,d)$ be a compact metric space and $\Phi:X\times \mathbb{R}\to X$ be a continuous flow on $X$.
They consider the ...
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0
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Intuition of the concepts behind Ergodic Theory
As a graduate math and physics student, I am introducing myself to the study of Ergodic Theory, reading Introduction to the Modern Theory of Dynamical Systems, by Katok and Hasselblatt. I understand ...
- 433
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0
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180
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Is there a general solution to $f(x)=f^{\circ n}(x)$?
This question has crossed my mind, and I tried finding some solutions to that functional equation, then to find a pattern.
It's surprisingly hard to find real functional equation calculators online, ...
- 333
3
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1
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98
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If $\omega_f(y) \subseteq \omega_f(x)$ and $int(\omega_f(y)) \neq \varnothing$ then $\omega_f(y) = \omega_f(x)$
We let $X$ be a compact metric space and $f:X\rightarrow X$ a continuous function. For any point $x$ we define the orbit of $x$ under $f$ as $orb_f(x) = \{f^n(x): n \in \mathbb{N}\}$, where $f^n$ is ...
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Has this metric been considered anywhere?
Let $X$ be a compact metric space and denote by $d$ the metric on $X$. I wondered whether the following metric $d_\infty : C(X,X)\times C(X,X) \rightarrow \mathbb{R_0^+}$ given by
$$d_\infty (f,g)= \...
- 161
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1
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There are dense orbits in the set of all allowed sequences
Lately, I have done an exercise in the book "Introduction to Dynamical Systems" by Brin and Stuck.
Exercise 1.4.5: Assume that all entries of some power of $A$ are positive. Show that in ...
- 794
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0
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Extending a covering map off an annulus
Let $R$ be a map from the Riemann sphere to itself, upon which its restriction to an annulus $A$ is a covering map to another annulus $B$.
Suppose there are critical points in one of the complementary ...
- 125
3
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0
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78
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Transitivity condition for monoid action
I am interested in a simple condition for a continuous monoid on a topological space to be topologically transitive. My setting is as follows:
Let $X$ be a $2$nd countable topological space and let $M$...
- 8,064
1
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1
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Critical points of rational mappings of annuli Fatou Component
Given a rational function $R$ from the Riemann sphere to itself and an annulus fatou component $A_0$ we can create the chain $$A_0 \xrightarrow{R} A_1 \xrightarrow{R} A_2 \xrightarrow{R} ...$$
One can ...
- 125
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1
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141
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Power of minimal system is transitive?
Let $(X,f)$ be a minimal dynamical system. What is the best thing we can deduce about $(X,f^n)$, some $n\in \mathbb{N}$? In particular, do we know that it is at least topologically transitive? Even ...
- 750
2
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1
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150
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Not every point is recurrent for a hyperbolic toral automorphism
Let $A: \mathbb{T}^2 \rightarrow \mathbb{T}^2 $ is a hyperbolic toral automorphism induced by an invertible integer matrix $A$ with no eigenvalues of modulus $1$, we say a point $x \in \mathbb{T}^2$ ...
- 987
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Uniquely Ergodic Transformation on compact groups
I am currently reading Einsiedler & Ward's book Ergodic Theory with a view towards Number Theory (GTM259) and am stuck at one of the exercises:
Exercise 4.3.2 (p.109): Let $T: X \to X$ be a ...
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How can I check that the following subshift of finite type is topologically mixing?
Let me consider the matrix
$$M=\begin{pmatrix}
0 & 1 & 0 & 0 \\
1 & 0 & 1 & 0 \\
0 & 1 & 0 & 1 \\
0 & 0 & 1 & 0
\end{pmatrix}.$$
I want to prove or ...
- 2,948
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Flows Overtime in Dynamical Systems
I am using the textbook Stability, Instability and Chaos by Glenninding to learn dynamical systems. The following excerpt is from the textbook's proof of Theorem 1.14:
(1.14) Theorem:
The set $\...
1
vote
1
answer
63
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Is there a standard term for the log cardinality, or entropy, of the pre-image of an element with respect to a function?
Suppose $h: X \to Y$ where $X$ and $Y$ are finite, i.e., $|X|, |Y| < \infty$. Is there a standard name for the quantity:
$$S_h(y) \equiv \log_2 |\text{Pre-image}_h(y)|?$$
For example, if the ...
- 141
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1
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69
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Inequality involving minimal cardinality of open subcovers
I am working on the following exercise, where I am not sure if the claim actually holds since I may have found a (simple) counter example. Maybe I am missing something?
Let $T \colon X \to X$ be a ...
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101
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Topological dynamical systems where all ergodic measures satisfy additional properties
Consider a topological dynamical system (tds) $(X, T)$, i.e. a compact metric space $X$ and a continuous map $T : X \to X$ (perhaps a homeomorphism). The ergodic probability measures for $T$ are ...
- 8,844
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What is the definition of the $\omega$-limit set of a point for semiflow?
Let $T$ be a topological semigroup, not necessarily discrete, and $\varphi:T\times X\to X$, where we denote by $(T, X) $, be semiflow on topological space $X$. This means that $\varphi$ is continuous ...
- 1,327
4
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2
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604
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Uniqueness of the Fixed Point when $f'(x) < 1$ at Fixed Points
For $f : [a,b] \rightarrow [a,b]$, show that if $f$ is continuous and differentiable and $f'(x) < 1$ at points such that $f(x) = x$, then $f$ has a unique fixed point.
The proof of the existence ...
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71
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Actions on Bernoulli space have almost the same orbits
Let $(\mathbb{B}, \nu)$ be the binary space with probability measure $\nu(0) = \nu(1) = \frac{1}{2}$. The map $T : \mathbb{B}^{\mathbb{N}} \to \mathbb{B}^{\mathbb{N}}$ is defined as left addition with ...
- 729
3
votes
3
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334
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Left Translation Action is Ergodic with respect to Haar
I am trying to solve exercises on ergodic group actions, from the A. Ioana's lecture notes "Orbit Equivalence of Ergodic Group Actions". The following exercise (p.3, Exr.1.14) has two parts, ...
- 729
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1
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352
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Prove that $\omega$-limit set of a recurrent point of a planar flow is a periodic orbit.
Let $f:U\rightarrow\mathbb{R}^2$ a $C^1$ vector field in an open set $U\subseteq\mathbb{R}^2$ and $p\in U$ a regular point of $f$. Show that if $p\in \omega_p(f)$, then $\omega_p(f)$ is a periodic ...
- 528
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107
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Every compact metric space contains some minimal set
A set $A$ is minimal if it is nonempty, closed, invariant (i.e. $f(A) \subseteq A$) and it does not contain any proper nonempty, closed, invariant subset.
Let $X$ be a compact metric space and $f:X \...
- 494
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The tent map system is transitive. Can we actually identify any of the infinitely many transitive points, however?
$\newcommand{\O}{\mathcal{O}}\newcommand{\G}{\mathcal{G}}\newcommand{\T}{\mathcal{T}}$TLDR; skip to the end of the preamble - we know that the tent map system is topologically transitive. However, do ...
- 44.7k
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1
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714
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Tent map is topologically transitive
Let $T:[0,1] \to [0,1]$ be the function $Tx= 2x$, if $ x \in [0, \frac{1}{2}]$ and $Tx = 2-2x$, if $ x \in ( \frac{1}{2} , 1] $. We say that a map is topologically transitive if, for any pair $U, V$ ...
- 71