Questions tagged [ds.dynamical-systems]
Dynamics of flows and maps (continuous and discrete time), including infinite-dimensional dynamics, Hamiltonian dynamics, ergodic theory.
2,482 questions
6
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Automorphism groups of subshifts and factor maps
Let $\pi : X \to Y$ be a factor map between subshifts over finite alphabets.
Let $\operatorname{Aut}(X)$ and $\operatorname{Aut}(Y)$ stand for automorphism groups of these shifts.
We say that $\varphi ...
- 1,658
4
votes
1
answer
235
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Dynamical analogue of Morse theory
Is there a Hamiltonian $H:\mathbb{R}^{2n} \to \mathbb{R}$ with the following property:
For two regular values $a<b$ for which $[a,b]$ consists of regular values, the dynamics of $X_H$ on $H^{...
- 356
2
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0
answers
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When is $f^*:T^*M\to T^*M$ an ergodic map for a diffeomorphism $f:M\to M$?
Let M be a differentiable manifold and $f:M \to M$ be a diffeomorphism. Then $f$ induces a natural map $f^* :T^*M \to T^*M$.
The pull back map $f^*$ is a symplectomorphism wrt the ...
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1
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Can the Reeb foliation of $S^3$ be realized as stable manifold foliation of a smooth hyperbolic discrete dynamic on $S^3$?
Can the Reeb foliation of $S^3$ be realized as foliation associated to stable(or unstable) manifolds of a hyperbolic discrete dynamic on $S^3$?If yes what is a precise formulation for that ...
- 356
2
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1
answer
139
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Can a chaotic trajectory solve an algebraic equation?
Given a polynomial ODE in $n$-dimensions of maximal degree $d$
$$\dot{x}_j=f_j(x)=\sum_{i_{1},\dots,i_{n}=1}^{d}a_{i_{1},\dots,i_{n}}^{j}x_{1}^{i_{1}}\dots x_{n}^{i_{n}} \quad \forall j=1,...,n$$
we ...
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1
vote
1
answer
154
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Nonamenable p.m.p. action on a standard probability space
Let $G$ be a discrete nonamenable countable group acting on a standard probability space $(X,\mu)$ through measure-preserving transformations.
Is the action of $G$ always amenable?
(Amenable action, ...
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20
votes
1
answer
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Roadmap to Ergodic Theory
I have recently been interested in going deeper into ergodic theory, beyond an introductory level of knowledge. Background wise, my training has mostly been in stochastic analysis, and I have a ...
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2
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1
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130
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Morse-Hedlund\Coven-Hedlund theorem for non-Abelian groups
There is a well know theorem by Coven and Hedlund, in Sequences with minimal block growth, stating that the complexity function of an aperiodic sequence\configuration $\omega\in \mathcal{A}^{\mathbb{Z}...
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2
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153
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Proof of Zimmer's cocycle super-rigidity theorem
I was reading the proof of Zimmer's cocycle super-rigidity theorem from the book 'Ergodic theory and semi-simple groups' by Robert Zimmer (Theorem 5.2.5, page 98). But I am not able to understand it. ...
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Pre-images of the critical point of $3.83 x(1-x)$
This question may be easy; however, I have been unable to locate any references regarding the specific scenario described below.
Let $T:[0,1]\to [0,1]$ be the quadratic map $T(x) = 3.83 x (1-x)$. It ...
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1
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0
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Periodic Orbit without Complex Eigenvalues
I am studying the following ODE system, representing a simple excitable circuit:
$$
\dot{V}_m = I_{app} - (V_m - \alpha_f PL(V_m) + \alpha_s PL(V_s))
$$
$$
\tau_s \dot{V}_s = V_m - V_s
$$
where
$$
PL(...
2
votes
2
answers
620
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A mutation of the Collatz disease
Given $k \in \mathbb N$, we define $f_k: \mathbb N \longrightarrow \mathbb N$ by
$$ f_k(x) = \begin{cases} \,\quad\dfrac{x}2 &\text{ if } x \text{ is even} \\\\ \dfrac{3x+3^k}{2} & \text{ if } ...
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11
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2
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What are the iterates of $x \mapsto 1 - \sqrt{1-x^2}$?
Let $f\colon [0,1]\to[0,1]$ be given by $f(x) = 1-\sqrt{1-x^2}$, i.e., the increasing auto-homeomorphism of $[0,1]$ whose graph is a quarter circle centered at $(0,1)$. I am interested in what can be ...
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2
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109
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Proving light escapes mirrors via ergodic theory of billiards
There's a longstanding open problem concerning whether or not it's possible to trap all the light from a point source using a finite collection of circles/lines whose sides are mirrors. This seems ...
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4
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0
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$f(n) = \frac{n^2 + n + 4}{2}$, $g(f(n)) = f(g(n))$ such that $g(n)$ is an integer
Let $n$ be a strict positive integer and let's define an integer sequence $f(n)$ :
$$f(n) = \frac{n^2 + n + 4}{2}$$
so
$$
\begin{split}
f (\Bbb N)& \triangleq {3,5,8,12,17,23,30,38,47,\ldots}\\
f(...
- 763
3
votes
1
answer
307
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"Ergodic theorem" for Markov kernels
Consider a discrete time Markov chain $(X_t)$ on a finite state space $\mathcal{S}$, with transition matrix $P$. Assume that the chain admits a stationary distribution $\pi$, which I will identify ...
1
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0
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94
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Convex combination of positive mean-ergodic operators
Let $T_1,T_2:L^1([0,1],\mathrm{d}x)\to L^1([0,1],\mathrm{d}x)$ be positive mean-ergodic operators such that:
For every $h:[0,1]\to \mathbb{R}_+$ we have that
$$\int_0^1 T_1 h(x)\mathrm{d}x = \int_0^1 ...
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21
votes
0
answers
416
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Can a 4D spacecraft, with just a single rigid thruster, achieve any rotational velocity?
(Copied from MSE. Offering four bounties over time, I got no response, other than twenty-nine upvotes.)
It seems preposterous at first glance. I just want to be sure. Even in 3D the behaviour of ...
- 281
2
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0
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313
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Correlation decay rate
Let $T$ be a continuous transformation of a probability measure space $(X,\mathcal{B}(X),\mu)$ and
$\varphi ,\phi \in L^2(\mu)$ (so-called observable) . The correlation function of $\varphi ,\phi$ (a ...
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5
votes
1
answer
265
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Equivalence of bounded gradient flow solutions and uniformly bounded gradient descent trajectories for definable functions
I am reading paper [1] by C. Josz regarding the global convergence of the gradient method. The main result is the following:
$\textbf{Theorem}$: For a definable differentiable function $f : \mathbb{R}^...
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1
vote
1
answer
74
views
Example of finite closed cover with entropy strictly greater than topological entropy
I'm reading "Topological entropy bounds measure-theorettic entropy", by L.W. Goodwyn.
enter link description here
After Proposition 2, he mentions that "finite closed cover can yield ...
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1
vote
1
answer
191
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Will this "tree" cover all rational numbers in a range?
Question
I am making a tree using the following two functions:
$$f(x)=\frac{x}{r},\quad g(x)=\frac{x+b}{r}$$
where $1<r<2$ and $0<b$ are rationals. Everything is a real number here.
The ...
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1
vote
2
answers
230
views
Hörmander’s propagation of singularities in two variables
I am trying to apply the propagation of singularities theorem to a distribution $u \in D’(M \times M)$ that verifies $Pu = f$, with $P$ a linear differential operator and $f \in D’(M \times M)$, as ...
- 319
1
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0
answers
93
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Time-inhomogeneous Krylov-Bogoliubov Existence Theorem
I am interested in what is known about the application of the Krylov-Bogoliubov existence theorem to the time-inhomogeneous case, especially as it relates to an underlying random dynamical system (...
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2
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0
answers
101
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Persistence of KAM tori as a function of dimension
I have tried posting this question in MSE, but I think it might be too technical so I'm trying again here.
In KAM theory one tries to describe the persistence of quasi-periodic motion when an ...
- 121
4
votes
1
answer
785
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The definition of simple eigenvalue
This question was posted a long time ago on the mathexchange, but I didn't get any answers there, and despite having discussed it with some colleagues, I don't think I have a definitive answer.
I am ...
- 136
0
votes
1
answer
198
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Invariance of the Kronecker factor
Let $(X,\mathcal{F},\mu,T)$ be a measure preserving system and $U_T$ Koopman operator on $L^2(X)$, i.e. $U_T f = f\circ T$. Note that, for the moment, I am not imposing any further assumptions on $X$, ...
- 141
1
vote
1
answer
235
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The liminf of an expression involving an irrational rotation
Let $0 < a < 1$ be an irrational number. Is it true that
$$\liminf_{n \in \mathbb N, n \to \infty} n \{na\} = 0?$$
Note: Here $\{\cdot\}$ denotes the fractional part.
- 6,195
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0
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61
views
Intuitive perspective on evolution of densities in dynamical systems
I am trying to understand the intuitive derivation of the Frobenius-Perron (FP) operator in the monograph:
Lasota, Andrzej, and Michael C. Mackey. Chaos, fractals, and noise: stochastic aspects of ...
2
votes
1
answer
359
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\...
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1
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168
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critical size of large systems of interacting particles for mean field approximation
When studying large systems of interacting particles, most of the work rely on mean field approximation by assuming that the number of particle goes to infinity. This allows to study the collective ...
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3
votes
1
answer
185
views
Is the weighted shift strong frequently hypercyclic?
One sided Shift
Let be $M$ separable metric space. Consider $X=M^{\mathbb{N}}$ the sequence space equipped with the product metric $d(x,y)=\sum_{i=1}^\infty |x_i-y_i|/2^i$ . Define the shift map $\...
- 757
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0
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146
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Linear dynamics in a function space
I posted the same question to Math Stackexchange earlier without much luck, so I am posting here.
I am dealing with a time-dependent model, which can be expressed as a function. $f$ is dependent on ...
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5
votes
1
answer
347
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Difference between the topological entropy and Hausdorff dimension for multifractal formalism
I have been reading some results about multifractal formalism. I noticed that some results were proved for the Hausdorff dimension and some results for the topological entropy (in the sense of Bowen). ...
- 1,043
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0
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67
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How to prove that pseudo entropy and topological entropy are the same with only Markov inequality and continuity?
Let $(X,\rho)$ be a compact metric space and $f:X\to X$ a homeomorphism. We say $(x_1,\ldots,x_{n})\in X^n$ is a partial $n$ orbit if $f(x_i)=x_{i+1}$. Let $Sep_{\epsilon}(X,\rho_n)$ be the maximal ...
1
vote
0
answers
2k
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Discrete dynamical system described by Dirichlet L-function using Yitang latest results on Landau–Siegel zero
Using the following definition of Dirichlet L-function
$$
L(1,\chi)=\begin{cases}
\dfrac{2\pi h}{w\sqrt{m}} & \textit{if}\ \chi(-1)=-1 \\\\
\dfrac{2 h \log{|\epsilon|}}{w\sqrt{m}} & \textit{...
9
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0
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259
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Centralizers in diffeomorphism groups
Let $M$ be a compact manifold and let $\mathrm{Diff}^{1}(M)$ denote the group of $C^{1}$-diffeomorphisms of $M$. Let $Z(f) := \{g \in \mathrm{Diff}^{1}(M) \mid gf = fg \}$ denote the centralizer of $f$...
3
votes
0
answers
71
views
Trapped vs. nonwandering points
For a continuous flow on a topological space the concept of a nonwandering point (in forward/backward time) is well-known. Another useful concept (for example from the point of view of scattering ...
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2
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1
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187
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How to find the hyperbolic dimension of map $f(z) = z^2$ of $\overline{\mathbb{C}}$ onto itself?
I am reading the research article "The Hausdorff dimension of the boundary of Mandelbrot set and Julia sets" by Shishikura. In his article, he defined hyperbolic sets and hyperbolic ...
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1
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1
answer
148
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Approximation of subshifts in Hausdorff distance
I have recently been interested in some questions which stem from taking subshifts which converge to a limiting subshift in the Hausdorff metric.
More specifically, given an alphabet $\mathcal{A}$, I ...
- 633
1
vote
1
answer
163
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Existence of center-stable manifold when the Jacobian is singular?
The following is a result from Shub's monograph "Global Stability of Dynamical Systems".
I dabble in the proof, and it appears to me that the existence of $W^{\rm cu}_{\rm loc}$ does not ...
- 511
3
votes
1
answer
140
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Time-dependent quantum dynamics as dynamical system and ergodic therorems
In full glory, a dynamical system is defined as a tuple $(T,M,\Phi)$ where $T$ is a monoid, written additively, $M$ is a set, and $\Phi$ is a function.
$$ \Phi : U \subset T \times M \to M$$
with
$ I(...
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2
votes
1
answer
142
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Ergodicity in irrational polygons and rational polyhedra
Steven Kerckhoff, Howard Masur and John Smillie in their paper [1] have proved that on rational polygons "For almost every direction the geodesic flow/billiard flow is ergodic". Is there any ...
- 41
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0
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83
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Question about stable manifold theorem and Frobenius integrability theorem
I have a question about Anosov diffeomorphism (Wikipedia: Anosov diffeomorphisms)
For hyperbolic fixed point $p$, $W^{s}(p)$ is a smooth manifold and its tangent space has the same dimension as the ...
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4
votes
1
answer
139
views
Finite number of ergodic random Dirac measures
Let $\Omega$ be a Polish locally compact space and $(\Omega, \mathscr{F}, \mathbb{P})$ be a probability space. Consider a measurable map
\begin{align*}
\theta\colon T\times \Omega &\to \Omega\\
(t,...
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8
votes
2
answers
340
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Does $x_0=1/3$ lead to periodicity in the logistic map $x_{k+1}=4x_k(1-x_k)$?
Does $x_0=1/3$ lead to periodicity in the logistic map $x_{k+1}=4x_k(1-x_k)$?
I believe it does not, but this is equivalent to proving that $(2\pi)^{-1}\arcsin(\sqrt{1/3})$ is irrational. I am ...
- 3,098
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votes
1
answer
321
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What is the logical complexity of the Mandelbrot Local Connectivity conjecture? (Is it equivalent to a statement of arithmetic?)
Denote by MLC the statement “the Mandelbrot set is locally connected” and MHC the statement “hyperbolic components are dense in the Mandelbrot set” (it is known that MLC implies MHC, and whether ...
- 32.5k
2
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0
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92
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A type of coupling problem II
This posting is related the following questions in MSE and in MO.
In general terms, suppose $(X,\mathscr{B},\mu)$ is a $\sigma$-finite measure space. If $\nu$ is another measure on $\mathscr{B}$, $\...
- 271
1
vote
1
answer
74
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Computing admissible patches of a substitution
I have been recently trying to look at substitution tilings with finite local complexity by examining their admissible patch\pattern atlas, which is sometimes called their language. I have also seen ...
- 633
3
votes
1
answer
170
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A type of coupling problem I
This posting is related to a recent question asked in MSE: Suppose $(X,\mathscr{B},\mu)$ is a $\sigma$-finite measure space. If $\nu$ is another measure on $\mathscr{B}$, $\nu(X)=\mu(X)$, and $\nu\ll\...
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