Questions tagged [ds.dynamical-systems]
Dynamics of flows and maps (continuous and discrete time), including infinite-dimensional dynamics, Hamiltonian dynamics, ergodic theory.
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Numerical detection of Cantori
It is known that as parameters vary in Hamiltonian system, KAM tori can break [1,2].
How to construct numerically the breaking tori?
The most relevant paper that I could find is [3,4].
But it uses ...
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3
votes
1
answer
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Extending isomorphism between subsystems in shift system
Let $(\Sigma^{\mathbb{Z}},S)$ be a left-shift system, where
$\Sigma$ is a metrizable compact set. Consider the automorphism group of it (bijective factor maps of itself), denoted by $G$.
Now let $(A,S)...
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A (possible) generic spectral property in one dimensional dynamics
Context and Definitions
Consider the interval $I=[0,1]$. We say that $T:I\to I$ satisfies the axiom A (I am following [1]) if:
$T$ has a finite number of hyperbolic periodic attractors; and
defining $...
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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{...
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4
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Deciding homomorphic images of De Bruijn graphs
The De Bruijn graph $B_n$ of
dimension $n$ (on the two-letter alphabet) is defined as the directed graph on
$2^n$ vertices and $2^{n+1}$ edges, where for every $w = w_0 \dots w_n \in
2^{n+1}$ we put ...
- 105
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87
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Aperiodic SFT equal to a substitution subshift
I was wondering whether there are primitive symbolic substitutions over $\mathbb{Z}^d$ and alphabet $\mathcal{A}$ whose associated subshift is equal to an aperiodic SFT. By SFT here I mean a subshift ...
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Maximal number of aperiodic Wang tiles
I was wondering whether there is an analogue result to the minimality of Wang tiling, in the direction of maximality.
I think that the paper by Jeandel and Rao, shows that the minimal number of Wang ...
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1
answer
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Ergodicity of linear dynamical systems and convergence of covariance matrices
Let $z(n+1)=Bz(n)+\xi(n+1)$ be an $N$-dimensional linear dynamical system with $\left(\xi(n)\right)_{n\in\mathbb{N}}$ being i.i.d. with $\xi(n)\sim\mathcal{N}(0,\Sigma_{\xi})$.
Assumptions: a) The ...
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the second largest eigenvalue of transfer operators
A Gauss map $T$ is mixing and satisfies Lasota-York inequalities. By Henon's theorem, we know that the transfer operator $\hat{T}$ associated with $T$ has a spectral gap. This means there exists a ...
user509763
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answer
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The current situation of the Godbillon-Vey invariant conjecture
Suppose we have a compact three-manifold $M$, a codimension-one foliation $F$ of $M$, and a one-form on $\alpha$, chosen so that $F$ is tangent to $\alpha = 0$. We deduce that $\alpha \wedge d\alpha = ...
- 312
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1
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A 1 dimensional foliation of $\mathbb{R}^4$ with few compact leaves
Inspired by An algebraic Hamiltonian vector field with a finite number of periodic orbits (2) we ask if there is a 1 dimensional analytic foliation of $\mathbb{R}^4$ which has at least 1 compact ...
- 312
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Analogue of Margulis height function in non lattice subgroups
I have been reading this paper https://link.springer.com/article/10.1007/s11854-017-0033-4 on singular system of linear forms and non escape of mass in homogeneous spaces $G/\Gamma$ where $ G=SL(m+n,\...
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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 ...
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1
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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^{...
- 312
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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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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 ...
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1
answer
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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$$
we ...
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Nonamenable p.m.p. action on a standard probability space
Let $G$ be a discrete nonamenable countable group acting on a standard probability space $(X,\mu)$ through measure-preserving transformations.
Is the action of $G$ always amenable?
(Amenable action, ...
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1
answer
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Roadmap to Ergodic Theory
I have recently been interested in going deeper into ergodic theory, beyond an introductory level of knowledge. Background wise, my training has mostly been in stochastic analysis, and I have a ...
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1
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Morse-Hedlund\Coven-Hedlund theorem for non-Abelian groups
There is a well know theorem by Coven and Hedlund, in Sequences with minimal block growth, stating that the complexity function of an aperiodic sequence\configuration $\omega\in \mathcal{A}^{\mathbb{Z}...
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Proof of Zimmer's cocycle super-rigidity theorem
I was reading the proof of Zimmer's cocycle super-rigidity theorem from the book 'Ergodic theory and semi-simple groups' by Robert Zimmer (Theorem 5.2.5, page 98). But I am not able to understand it. ...
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Pre-images of the critical point of $3.83 x(1-x)$
This question may be easy; however, I have been unable to locate any references regarding the specific scenario described below.
Let $T:[0,1]\to [0,1]$ be the quadratic map $T(x) = 3.83 x (1-x)$. It ...
- 233
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answers
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Periodic Orbit without Complex Eigenvalues
I am studying the following ODE system, representing a simple excitable circuit:
$$
\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(...
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2
answers
561
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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
$$ f_k(x) = \begin{cases} \,\quad\dfrac{x}2 &\text{ if } x \text{ is even} \\\\ \dfrac{3x+3^k}{2} & \text{ if } ...
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What are the iterates of $x \mapsto 1 - \sqrt{1-x^2}$?
Let $f\colon [0,1]\to[0,1]$ be given by $f(x) = 1-\sqrt{1-x^2}$, i.e., the increasing auto-homeomorphism of $[0,1]$ whose graph is a quarter circle centered at $(0,1)$. I am interested in what can be ...
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Proving light escapes mirrors via ergodic theory of billiards
There's a longstanding open problem concerning whether or not it's possible to trap all the light from a point source using a finite collection of circles/lines whose sides are mirrors. This seems ...
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Purely non-atomic measure on the Gromov boundary of a finitely generated free group
In the set-up of my previous post, let $\theta$ be a purely non-atomic probability regular measure defined on the Borel $\sigma$-algebra of the metric space $(\partial F, d)$. We say $\theta$ admits a ...
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$f(n) = \frac{n^2 + n + 4}{2}$, $g(f(n)) = f(g(n))$ such that $g(n)$ is an integer
Let $n$ be a strict positive integer and let's define an integer sequence $f(n)$ :
$$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(...
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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 ...
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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
$$\int_0^1 T_1 h(x)\mathrm{d}x = \int_0^1 ...
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Can a 4D spacecraft, with just a single rigid thruster, achieve any rotational velocity?
(Copied from MSE. Offering four bounties over time, I got no response, other than twenty-nine upvotes.)
It seems preposterous at first glance. I just want to be sure. Even in 3D the behaviour of ...
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answers
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Correlation decay rate
Let $T$ be a continuous transformation of a probability measure space $(X,\mathcal{B}(X),\mu)$ and
$\varphi ,\phi \in L^2(\mu)$ (so-called observable) . The correlation function of $\varphi ,\phi$ (a ...
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Equivalence of bounded gradient flow solutions and uniformly bounded gradient descent trajectories for definable functions
I am reading paper [1] by C. Josz regarding the global convergence of the gradient method. The main result is the following:
$\textbf{Theorem}$: For a definable differentiable function $f : \mathbb{R}^...
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vote
1
answer
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Example of finite closed cover with entropy strictly greater than topological entropy
I'm reading "Topological entropy bounds measure-theorettic entropy", by L.W. Goodwyn.
enter link description here
After Proposition 2, he mentions that "finite closed cover can yield ...
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answers
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dynamical system modified by PD matrix
Assume we have a dynamical system, i.e an ODE of the form
$$
\frac{d\vec{x}}{dt}=g(\vec{x})
$$
which we know how to solve.
Now consider a modified ODE
$$
\frac{d\vec{x}}{dt}=\hat{A}g(\vec{x})
$$
with $...
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1
answer
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Will this "tree" cover all rational numbers in a range?
Question
I am making a tree using the following two functions:
$$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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vote
2
answers
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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 ...
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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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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 ...
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votes
1
answer
564
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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 ...
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answer
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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$, ...
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Local complexity of tilings under substitutions
I am trying to read this survey of E. Arthur Robinson, about tilings of $\mathbb{R}^d$. I have some familiarity with 'symbolic' tilings, but I don't think I have a good intuition on 'geometric' ...
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answer
228
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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.
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61
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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
253
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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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votes
1
answer
152
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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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votes
1
answer
179
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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 $\...
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145
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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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votes
1
answer
255
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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). ...
- 1,031
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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 ...
