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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49 views

Non-compact dynamical systems

In topological dynamics, most of the time, we consider the continuous action of a (semi)group $G$ on a compact Hausdorff space $X$. In this context, we can envelop the group in a compact left ...
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Smoothening pseudo-Anosov flows

A topological Anosov flow on a closed 3-manifold can be replaced by a smooth Anosov flow using an argument of Fried: use Markov partitions to find a surface of section, put in other terms, one can ...
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Is my ansatz for finding $n$-periodic-points of the exponential-function exhaustive?

In the problem of finding fixed and periodic points of the complex exponential-function I introduced the notation for the function of a vectorial argument $K$ $$ T_n=\text{Find}(K) \qquad \text{...
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If $(Y,T)$ is a connected minimal system with a symbolic extension of linear word complexity, is $(Y,T)$ equicontinuous?

Let $(Y,S)$ be a minimal topological dynamical system such that $Y$ is connected. A simple example of a system like this is an irrational rotation of the circle, and it is known that Sturmian ...
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41 views

On smooth extensions of functions

Let $f(x) = \left(I - \hat{n}\hat{n}^T \cdot\textbf{1}_{\vec{n}^TAx \geq 0}\right)Ax$, where $I$ is the identity matrix, $A$ is a (symmetric) $d\times d$ positive definite matrix, $\hat{n}$ is an ...
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1answer
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is the Lyapunov exponent a continuous function of the invariant measure w.r.t weak-* topology?

I hope for some relerant results for the following question: Is the lyapunov exponent continuous with respect to the measure? Assume $M$ is a manifold, $f$ is a diffeomorphism on $M$, $m$ is an ...
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2answers
72 views

Transverse invariant measures to vector fields

Given a smooth, non-vanishing vector field on a compact manifold, when does the 1-dimensional foliation given by its integral curves admit a transverse invariant measure? I've seen examples of higher-...
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56 views

Counting simple closed curves

I'm currently trying to understand how to count simple closed curves. I've been reading Alex Wright's survey (https://arxiv.org/pdf/1905.01753.pdf). However, I don't feel like I'm getting the big ...
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32 views

Center-stable manifold theorem on manifold with boundary

I would like to see if there is a Center-stable manifold theorem on the phase space that is a manifold with boundary. Suppose $f:M\rightarrow M$ is a diffeomorphism, according to Theorem III.7 in "...
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1answer
30 views

Difference of hypercyclic operator and identity

Let $B$ be a separable Banach space, $L:B \to B$ be a hypercyclic operator, $k>0$, $I_B$ the identity on $B$, and define $L_k: =k (I_B - L)$. When is $L_k$ hypercyclic on $B$? Can anything else ...
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+50

Question on “semi-linear” dynamical systems

I am interested in understanding the convergence properties of a dynamical system at zero. We know that a dynamical system of the form $x_{k+1} = Ax_{k}$ where $A$ is a symmetric positive definite ...
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Convergence to equilibrium of a nonlinear dynamical system

Consider the following dynamical system in $\mathbb{R}^n$ $$ \dot{x} = -x + A\tanh(x)=:f(x) $$ where $x = (x_1,...,x_n) \in \mathbb{R}^n$, $A$ is a real matrix with spectral radius $\rho(A) < 1$, ...
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Floquet theory and Poincaré theorem on the continuation of periodic orbit

I read about the Floquet theory and a theorem that it named Poincaré's theorem of the continuation of periodic orbit. Poincaré's Theorem: Consider a dynamical system depending on the parameter $\...
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2answers
104 views

Does this strong form of being almost 1-to-1 imply injectivity?

Let $\pi\colon(X,T)\to (Y,T)$ be a factor map between minimal subshifts. Suppose there exists $\tilde{Y} \subseteq Y$ such that $\# \pi^{-1}(y) = 1$ for all $y \in \tilde{Y}$. $\tilde{Y}$ is a ...
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1answer
113 views

Properties of the spectrum of the Koopman representation

Let $G$ be a discrete countable infinite group acting on a compact metric space $X$ via homeomorphisms preserving a probability measure $\mu$. A function $\lambda\colon G\to \mathbb C$ is an ...
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1answer
34 views

Action of a toral automorphism on a Markov partition

Let $$A = \begin{bmatrix} 1 & 1 \\ 1 & 0\\ \end{bmatrix}.$$ Then the eigenvalues of $A$ are $1/2(1+\sqrt{5})$ and $1/2(1-\sqrt{5})$. The eigenvector corresponding to the unstable eigenvalue ...
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Determining the behavior of a contraction mapping with undefined points

Label $X$ as the real interval $[0, a]$ where $a \in \mathbb{R}^+$, so that $\text{int}(X) = (0, a)$ labels the interior of $X$ and $\partial X$ labels the boundary of $X$. I have a function $f:\text{...
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Exponential iterates of a complex number

Let $f:\mathbb C\to \mathbb C$ be defined by $f(z)=e^z-1$. Let $f^n$ denote the $n$-fold composition of $f$. In my new paper Erdős space in Julia sets I show that $$Z:=\{z\in \mathbb C:\lvert\...
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Results on: (path/initial condition)-dependent variant of exponential map generates compactly supported diffeomorphisms

Let $M$ be a connected and simply connected Riemannian manifold, non-compact, and suppose that $\{V_p\}_{p \in M}$ is a family of vector fields on $M$ indexed by $M$. Suppose moreover that the map $$ ...
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118 views

Central limit theorem versus entropy in dynamical systems context

A dynamical system $(S^1,T, \mu)$, $T_* \mu=\mu$, $T$ ergodic, $S^1$ is circle. Assume it has central limit theorem. Want to know the relation between its measure-theoretic entropy $h_{\mu}(T)$ and ...
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Lyapunov indices of a product of operators

The deterministic part of the proof of the multiplicative ergodic theorem can be proven using Proposition 1.3 in the paper Lyapunov indices of a product of random matrices.$^1$ They consider a ...
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1answer
90 views

Physical measures that are not SRB

It is quite easy to construct a dynamical system which has a physical measure with a positive Lyapunov exponent and zero entropy, just a figure $\infty$ system. By Pesin's entropy formula such a ...
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2answers
473 views

Monotonic and bounded sequences throughout mathematics [closed]

When I refer to the Monotone Convergence Theorem below, I refer to the very simple claim that if a non-decreasing sequence has an upper bound then it converges. I don't refer to the claim from Measure ...
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2answers
168 views

Seeking to understand meaning of “von Neumann spectrum” in a paper of Bader–Furman–Shaker

In attempting to understand the paper "Superrigidity, Weyl groups, and actions on the circle" of Uri Bader, Alex Furman and Ali Shaker (linked at Furman's page) I find that towards the end of the ...
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Stability of a continuous piecewise linear map

I am studying random perturbation of a system that is continuous and piecewise linear. More precisely: I am given a map $\Phi_1:\mathbb{R}^d\to \mathbb{R}^d$ such that $$ \Phi_1(x) = \left\{\begin{...
4
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1answer
105 views

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 ...
5
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1answer
192 views

Metrics on torus without closed contractible geodesics

It is easy to see that any closed geodesic on a flat 2-torus is noncontractible. Further the same holds true for a torus of revolution. Indeed either a closed geodesic is a meridian and therefore ...
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2answers
239 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 $\mathbb{R}$?
2
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1answer
138 views

A subadditive maximal ergodic theorem

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $\tau:\Omega\to\Omega$ be a measurable map on $(\Omega,\mathcal A)$ with $\operatorname P\circ\:\tau^{-1}=\operatorname P$, $Y_n:\...
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2answers
463 views

Do infinitely nested radicals have any applications?

There is a simple necessary and sufficient condition for a continued radical of the form $\sqrt{a_1 + \sqrt{a_2 + \dotsc}}$ to converge (where all terms $a_1, a_2$ etc. are nonnegative). Namely, that ...
2
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1answer
198 views

A second order nonlinear ODE

In my research (in differential geometry) I recently came across the following nonlinear second order ode: $$\frac{f''(x)}{f'(x)}-\frac{2}{x}+\frac{f'(x)+1}{2f(x)-x-1}+\frac{f'(x)-1}{2f(x)+x}=0$$ It ...
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0answers
40 views

Alternate characterization of floquet multipliers: Floquet theory

Given an autonomous ode $\dot{x}=f(x)$ in $\mathbb{R}^n$ possessing a period-p time-periodic solution $\bar x(t)$, one can use the so-called variational equation about $\bar x$ to study its stability. ...
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4answers
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For $x$ irrational, is $a_{n} =\sum_{k=1}^{n}(-1)^{⌊kx⌋}$ unbounded?

For $x$ irrational, define $a_{n} :=\sum_{k=1}^{n}(-1)^{⌊kx⌋}$. Can you prove that $\left\{a_n\right\}$ is unbounded? I feel that it is not easy to treat every irrational $x$. I have asked in S.E. ...
3
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1answer
80 views

Almost one-to-one endomorphism of minimal subshift?

Let $(X,T)$ be a minimal subshift. Can it happen that an endomorphism $\varphi\colon (X,T) \to (X,T)$ is almost 1-to-1 but not 1-to-1? Can it happen that a factor $\pi\colon (X,T) \to (Y,T)$ between ...
4
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1answer
110 views

Leaves of stable foliation of holomorphic Anosov diffeomorphism

I'm trying to understand the first half of the paper "Holomorphic Anosov systems" by E. Ghys (the journal reference is Inventiones mathematicae volume 119, pages 585–614(1995)). My question is about a ...
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70 views

Factor map between subshifts preserving topological pressure (or measure-theoretic entropy)

Let $G$ be a countable amenable group and let $X,Y$ be subshifts with finite alphabet over $G$. Suppose that $h(X) = h(Y)$ (equal topological entropy). I am interested in continuous factor maps $\pi: ...
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97 views

Finding invariant closed subspace which are also subgroups for the action of $\text{SL}(2, \Bbb Z)$ on $\Bbb R^n\times \Bbb R^n$

I recently came across to the following action of $\text{SL}(2,\Bbb Z)$ on the space $\Bbb R^n\times\Bbb R^n$ defined as $$ \begin{pmatrix} a & b \\ c & d \end{pmatrix}\cdot \big(v,\,w\big)\...
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0answers
59 views

Bound on number of linearly independent eigenvectors of adjoint of composition operator

Fix $N>1$. Let $f\in C^{\infty}(\mathbb{R},\mathbb{R})$ be such that the composition operator via $$ \begin{aligned} C_f:C^{\infty}(\mathbb{R},\mathbb{R}) &\rightarrow C^{\infty}(\mathbb{R},\...
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0answers
49 views

Dense stratification of a separable Hilbert space

Let $\{X_i\}_{i \in \mathbb{N}} $ be a sequence of $n$-dimensional linear subspaces of the separable Hilbert space $H$ and let $\{\phi_i\}_{i \in I}$ be a sequence of continuous injective linear maps ...
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46 views

SRB measures on hyperbolic sets of random dynamical systems

It is well-known that a $C^2$ Axiom A basic set $\Lambda$ is an attractor if and only if the topological pressure with respect to the negative unstable log-determinant vanishes (see Theorem 4.11 in ...
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19 views

Lower bounds of kappa class functions

I saw in the paper "Smooth Satabilization Implies Coprime Factorization" of Eduardo Sontag the following argument: Given a smooth map $a:\mathbb{R} ^{n}\rightarrow\mathbb{R}^{+}$, let $\rho$ be any ...
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1answer
84 views

Consequences of invariant-subspace problem to Li–Yorke chaos [closed]

The invariant-subspace problem is probably an open problem for reflexive spaces which asks: Does every bounded linear operator on an infinite-dimensional separable Hilbert space have a non-trivial ...
2
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1answer
170 views

Seeking a Lyapunov function for a SIR model with immunity loss

We add the immunity loss to the SIR model and obtain the following autonomous system. $$ \begin{align} s' &= -is+\alpha r \\ i' &= i s - \gamma i\\ r' &= \gamma i-\alpha r \end{align} \...
4
votes
2answers
77 views

Minimal subshift with some $x \in X$ such that $x_{(-\infty,0)}.x_0x_0x_{(0,\infty)} \in X$?

There exists a minimal subshift $X$ with a point $x \in X$ such that $x_{(-\infty,0)}.x_0x_0x_{(0,\infty)} \in X$?
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51 views

Elliptic foliations of the plane

A $1$ dimensional foliation of the plane $\mathbb{R}^2$is called elliptic if it admits a non vanishing smooth tangent vector field $X$ with the following properties: The differential operator ...
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0answers
97 views

A Fourier elliptic vector field on a Riemannian manifold

Motivation for this question: Let $X$ be a vector field on a manifold $M$. Obviously the differential operator $D:C^{\infty}(M)\to C^{\infty}(M)$ with $D(f)=X.f$ is not an elliptic opetator when $\...
2
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1answer
92 views

A reference to the fact that a topologically transitive action of a group on a compact metrizable space has a dense orbit

I need a proper reference to the following obvious fact: An action of a group $G$ on a nonempty compact metrizable space $K$ is topologically transitive (= the orbit $GU$ of any nonempty open set $...
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0answers
80 views

Existence of large first return times

Let $(X,T,\mu)$ be a measure preserving system, with $\mu$ a probability measure. Let $E \subset X$ of positive measure and $\tau_E$ be the first return time to $E$. Then the Kac Lemma asserts that $$\...
1
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1answer
101 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 ...
5
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1answer
169 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 ...

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