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.
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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 ...
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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,...
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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 ...
- 774
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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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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}\...
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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 ...
- 774
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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
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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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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 ...
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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, ...
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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)= \...
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