All Questions
Tagged with gn.general-topology ds.dynamical-systems
81
questions
1
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0
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67
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Approximating evalutation maps at open sets over invariant measures
Let $G$ be a group acting by homeomorphisms on a compact metrizable space, say $X$; let's denote by $\alpha:G\to\mathrm{Homeo}(X)$ the action, $g\mapsto\alpha_g$, and consider the weak-$^*$ compact ...
- 93
3
votes
0
answers
85
views
Progess on conjectures of Palis
I came across a "A Global Perspective for Non-Conservative Dynamics" by Palis. He has some conjectures
"Global Conjecture:
There is a dense set $D$ of dynamics such that any element of ...
- 227
0
votes
0
answers
37
views
Generic non-existence of 1. Integral of continuous DS
Let $M$ be a compact manfiold and $F \in \mathcal{X}(M)$. We define a DS on $M$ by
$$\dot{\mathbf{x}}=F(\mathbf{x}(t))$$
In 1 it was shown by Hurley, that a generic diffeomorphism on $M$ does not have ...
- 227
0
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0
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290
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Proof that a first integral is not a constant function
Let $U$ be an (open) set in $\mathbb{R}^n$. And we are given a set of $m$ basis functions
$$B=\{\psi_i(x): U \rightarrow \mathbb{R}\mid i=1,\ldots,m \}$$
such that all of them are differentiable and ...
- 227
0
votes
0
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69
views
Example of DS with a dense trajectory in the whole state space
Let $U \subset \mathbb{R}^n$ be an open and connected set. We assume there is a vector field $F \in \mathcal{C}^1(\overline{U})$ giving rise to a DS ($\overline{U}$ denotes the closure)
$$\dot{\mathbf{...
- 227
2
votes
1
answer
199
views
Equivalence of the definitions of exactness and mixing
Let $f:X \to X$ be a continuous map, where $X$ is a compact metric space. We say that $f$ is (locally) expanding if there are constants $\lambda >1$ and $\delta_0 > 0$ such that, for all $x, y\...
- 118
8
votes
1
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269
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State of the art on: "If a dynamical system is Li-Yorke chaotic, does there exist a Cantor scrambled set?"
The following problem is presented in the paper Recent development of chaos theory in topological dynamics - by Jian Li and Xiangdong Ye :
"If a dynamical system [$(X,f)$, $X$ metric space, $f$ ...
- 149
6
votes
2
answers
426
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What is the name for a point that is periodic to within $\varepsilon$?
Let $X$ be a set and $f: X \to X$ a function. A point $x \in X$ is, of course, said to be periodic for $f$ if $x \in \{f(x), f^2(x), \ldots\}$.
Now suppose that $X$ is a topological space and $f$ is ...
- 27k
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106
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Polynomial entropy of topological dynamical systems
For a continuous mapping $\varphi: X \to X$ on a compact Hausdorff space $X$ we define the entropy as follows:
Given open covering $\mathcal{U}$, $\mathcal{V}$ of $X$, we call $\mathcal{V}$ a ...
- 359
9
votes
1
answer
297
views
Equivalent definitions of topological weak mixing
A dynamical system $f:X\to X$ is said to be topologically transitive if for any two nonempty open sets $U,V$ there exists $n \in \mathbb{Z}$ such that $f^{\circ n}(U) \cap V \neq \emptyset$. The ...
- 91
5
votes
1
answer
192
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The cardinal characteristic $\mathfrak r_{(X,f)}$ of a dynamical system
I am interested in a "dynamical" modification of the cardinals $\mathfrak r$ and $\mathfrak r_\sigma$, well-known in the theory of cardinal characteristics of the continuum.
For a compact ...
- 40.2k
0
votes
0
answers
64
views
Li-Yorke sensitivity Vs Li-Yorke dense chaos
Let $X$ be a compact metric space, $X*X$ its cartesian product, and $A$ a subset of $X*X$.
Are the following two properties the same, or e.g. one is stronger than the other?
$A$ is dense and residual ...
- 131
0
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0
answers
63
views
a lemma on interval translation map
Consider the map $S:T^1 \to T^1$ where $x \mapsto x+c_j$ , mod 1 where $c_j$'s are real numbers. We represent $T^1$ as a union of disjoint subsegments $M_j=[t_j,t_{j+1})$, $j=0 , \cdots ,n , t_0=t_n$....
- 145
4
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1
answer
477
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Are these topological sequence entropy definition equivalent?
I am working on Möbius disjointness for models of topological dynamic systems. In that purpose, I try to understand the notion of topological entropy. We know, for a t.d.s $(X,T)$ that it is defined ...
- 137
1
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0
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179
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Subset of the domain of attraction
Let $x \in R^n$ and $f : R^n \to R^n$, $f\in C^1$
$$
\frac{\mathrm{d}}{\mathrm{d}t} x(t) = f(x(t))
$$
be such that $f(0) = 0$ is asymptotically stable. The domain of attraction is the set of initial ...
- 61
3
votes
2
answers
253
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Points attracting to 0 are dense in $\mathbb C$
I know that the following proposition is true, but at the moment I can't see how to prove it.
Define $f(z)=e^z-1$ for all $z\in \mathbb C$. Then $A:=\{z\in \mathbb C:f^n(z)\to 0\}$ is dense in $\...
- 2,993
2
votes
2
answers
288
views
Construct a homeomorphism whose periodic points set is not closed
I'm looking for a simple example in discrete dynamical systems whose periodic points set is not necessary closed.
I've seen some example in websites but they are not that simple and discrete.
Note ...
- 145
1
vote
1
answer
441
views
Mandelbrot set and logistic map connection
I'm currently writing an undergraduate thesis on chaos theory with a particular focus on the connection between the Mandelbrot set and the logistic map. I have found scattered posts on this site, ...
10
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3
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534
views
Can an "almost injective'' function exist between compact connected metric spaces?
Let $\pi: X\to Y$ be a surjective continuous function between the compact, metric and connected spaces $X$, $Y$, and $Y_0 = \{y\in Y: \#\pi^{-1}(y)>1\}$. Suppose that:
$Y_0$ is dense in $Y$,
$Y\...
- 165
5
votes
2
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154
views
Local cross-sections for free actions of finite groups
Let $G$ be a finite group, let $X$ be a locally compact Hausdorff space, and let $G$ act freely on $X$. It is well-known that the canonical quotient map $\pi\colon X\to X/G$ onto the orbit space $X/G$ ...
- 717
4
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1
answer
308
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Let $U$ be a simply connected open subset of ${\Bbb S}^2$, is the complement of $U$ also simply connected?
I was looking into particular cases for the Poincaré-Bendixson theorem and I came across a topological problem about simply connectivity.
If $\gamma$ is a Jordan curve in ${\Bbb S}^2$ then using ...
- 452
18
votes
2
answers
1k
views
Existence of continuous map on real numbers with dense orbit?
Does there exist a continuous map $f:\mathbb{R}\rightarrow \mathbb{R}$ such that the forward orbit of 0 is dense in $\mathbb{R}$?
- 179
1
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1
answer
130
views
A permutation group inducing a topologically transitive action without dense orbits on $\omega^*$
Let $G$ be a subgroup of the permutation group $S_\omega$ of the countable infinite set $\omega$. Each bijection $g\in G$ admits a unique extension to a homeomorphism $\bar g$ of the Stone-Cech ...
- 40.2k
5
votes
1
answer
437
views
A topologically transitive dynamical system without dense orbits
By a dynamical system I understand a pair $(K,G)$ consisting a compact Hausdorff space and a subgroup $G$ of the homeomorphism group of $K$.
We say that a dynamical system $(K,G)$
$\bullet$ is ...
- 40.2k
7
votes
0
answers
264
views
Possible Birkhoff spectra for irrational rotations
Let $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ be the unit circle (think of it as of the interval $[0,1)$ with endpoints identified). Assume that $\alpha$ is irrational and consider the rotation by $\alpha$, ...
- 1,578
15
votes
1
answer
474
views
Group actions and "transfinite dynamics"
$\DeclareMathOperator\Sym{Sym}$I have a question about what I shall name here "transfinite dynamics" because it involves iterating a topological dynamical system $G \curvearrowright X$ ...
- 4,010
1
vote
1
answer
125
views
Is the set of non-escaping points in a Julia set always totally disconnected?
I am looking for examples of transcendental entire functions $f:\mathbb C\to \mathbb C$ such that the set of non-escaping points in the Julia set of $f$ is not totally disconnected. I denote this set $...
- 2,993
3
votes
1
answer
164
views
Reversal of open cover with topologically transitive dynamical system
Let $X$ be a separable metric space, $\phi\in C(X,X)$ be a topologically transitive dynamical system, and $V$ be a non-empty open subset of $X$, and $\nu$ be a locally-positive and atomless Borel ...
- 5,001
0
votes
1
answer
88
views
Topologically transitive dynamical system mapping space into ball
Let $X$ be a separable Hausdorff topological space and $\phi \in C(X,X)$ be a topologically transitive map. Further, let $V$ be a fixed non-empty open subset of $X$. Then does there necessarily ...
- 5,001
6
votes
2
answers
388
views
Do mixing homeomorphisms on continua have positive entropy?
I am trying to understand relations between various measures of topological complexity. I have read that expansive homeomorphisms on continua, for example, have positive entropy. But I do not know ...
- 2,993
2
votes
1
answer
116
views
Size of the orbit of a dense set
This question is a follow-up to: this post.
Let $X$ be a separable Banach space, $\phi\in C(X;X)$ be an injective continuous non-affine map, and $A$ be a dense $G_{\delta}$ subset of $X$. How big ...
- 63
13
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1
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450
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Is the set of escaping endpoints for $e^z-2$ completely metrizable?
Let $f:\mathbb C \to \mathbb C$ be the complex exponential $$f(z)=e^z-2.$$ It is known that $J(f)$, the Julia set of $f$, is a uncountable collection of disjoint rays (one-to-one continuous images ...
- 2,993
1
vote
1
answer
156
views
Is the infinite product map $(∏_{i=1}^{∞}S_{i})×f$ topologically transitive
Let $f:X→X$ be a map. We say that $f$ is topologically mixing if for every open subsets $U,V$ of $X$, there exists $N$ such that for every $n≥N$ the set $f^{n}(U)∩V$ is non-empty.
Let $S : X → X$ and ...
- 963
2
votes
1
answer
118
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Irreducible subcontinuum of Lorenz attractor?
In my first question Lorenz attractor path-connected?, some are saying the Lorenz attractor $\mathscr L$ is not path-connected.
But suppose $x$ and $y$ are two points in different path components of ...
- 333
9
votes
1
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751
views
Lorenz attractor path-connected?
Can we tell if the Lorenz attractor is path-connected? By the attractor I do not mean only the line weaving around, but rather its closure.
EDIT: The answer below is unsatisfactory, and possibly ...
- 333
11
votes
1
answer
467
views
Do solenoids embed into Möbius strips?
I found a strange attractor which looks a lot like a solenoid. The attractor continuum is the closure of a continuous line which limits onto itself, and it is locally homeomorphic to Cantor set times ...
- 945
1
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0
answers
124
views
How many two-dimensional space filling Hilbert-like curves are there?
I'm interested in filling 2d square with space filling, non-self-intersecting, locality preserving, self-similar curves, like Hilbert curve. I found interesting work concerning three dimensional case ...
- 355
4
votes
2
answers
221
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Inverse image of rational values
I am a postgraduate student of physics. While doing some research on Poincare's work on the integrability of the three body problem, I came up with the following problem (which I feel unable to handle,...
- 41
10
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1
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202
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homeomorphisms induced by composant rotations in the solenoid
Let $S$ be the dyadic solenoid.
Let $x\in S$, and let $X$ be the union of all arcs (homeomorphic copies of $[0,1]$) in $S$ containing $x$.
$X$ is called a composant of $S$.
It is well-known ...
- 945
6
votes
1
answer
413
views
Transitive homeomorphisms of Erdős spaces
A surjective homeomorphism $h:X\to X$ is minimal if $$\overline{\{h^n(x):n\in \mathbb N\}}=X$$ for every $x\in X$. In other words, the orbit of each point is dense.
Does either of the Erdös spaces $\...
- 2,993
4
votes
1
answer
146
views
Equicontinuity and orbits of compact open sets
Let $X$ be a compact zero-dimensional space, let $S \subseteq \mathrm{Homeo}(X)$ and let $U$ be a compact open subset of $X$. Suppose that $s^{-1} \in S$ for all $s \in S$, and that $S$ restricts to ...
- 4,678
3
votes
2
answers
296
views
Inverse limit space and $C^1$ topology
Let $f_n$ be a sequence of $C^1$-maps on closed manifold $M$. If $f_n \to f$ in $C^1$-topology. Does $M_{f_n}$ converges to $M_f$ in the Hausdorff distance?
We define $M(f)=\{\bar{x}=(x_j) \in M^{\...
11
votes
0
answers
212
views
Shift invariant measurable selection theorem
Let $(X,\mathcal{F})$ be some measure space and endow $\mathbb{R}^\mathbb{Z}$ with the product topology and borel $\sigma$-field. Let $F$ be a point to set mapping $X^\mathbb{Z}\rightarrow \mathcal{P}(...
- 479
2
votes
1
answer
517
views
Is there a minimal, topologically mixing but not positively expansive dynamical system?
Is there a compact metric space $X$ and a function $f:X\to X$ such that the dynamical system $(X, f)$ has the following three properties?
minimal
topologically mixing (a map $f$ is topologically ...
- 83
6
votes
2
answers
581
views
Is there a topologically mixing and minimal homeomorphism on the circle (or on $\mathbb S^2$)?
The irrational rotation on the circle is both a homeomorphism and minimal but is not topologically mixing. The argument-doubling transformation on the circle is topologically mixing but is neither a ...
- 83
1
vote
0
answers
131
views
Inverse limits of the interval with a single bonding map below the identity
My question is as follows.
QUESTION. Is there a topological description of the class of arc-like continua that arise as inverse limits of $[0,1]$ with a single continuous surjective bonding map $f\...
- 6,390
2
votes
1
answer
356
views
Applications of topology to discrete dynamical systems?
I'd like to know some of the applications of topology to discrete dynamics. By discrete dynamics I loosely mean studying maps between discrete sets.
I mean cases where adding a topology to the sets ...
- 803
1
vote
1
answer
183
views
Identifying attractors in high dimensional dynamical sytems [closed]
I have a high dimensional dynamical system, and I was wondering if there is a method to identify the various attractors of the system i.e, a way of mapping the energy landscape?
I was thinking of a ...
- 19
6
votes
0
answers
352
views
Topologically transitive, pointwise minimal systems
I'm cross-posting this from SE.
Let $T$ be a group, and let $(X,T)$ be a flow, i.e. $X$ is a compact Hausdorff space and $T$ acts on $X$ by homeomorphisms. A flow $X$ is called topologically ...
- 369
7
votes
1
answer
369
views
Approximation of topological dynamical systems?
I'm trying to find references to approximations of topological dynamical systems in the following sense:
A topological dynamical system $(X, f)$ consists of a topological space (typically compact ...
- 141