# Questions tagged [ds.dynamical-systems]

Dynamics of flows and maps (continuous and discrete time), including infinite-dimensional dynamics, Hamiltonian dynamics, ergodic theory.

1,577 questions
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### Clarification needed on vector field conditions in Smale's “On gradient dynamical systems”

I previously posted the question on MSE but I haven't received an answer. I'm now posting it here in a slightly revised form. In S. Smale's, “On gradient dynamical systems,” Ann. of Math. (2), vol. ...
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### Updated background on Hilbert 16th problem?

What is the current situation of the second part of the Hilbert 16th problem? What are the most updated news on this problem?
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### Asymptotically invariant maps and strongly ergodic actions

Let $\Gamma$ be a countable group which acts strongly ergodically on a probability measure space $(X,\mu)$. Let $\sigma_k:X \rightarrow Y$ be a sequence of measurable functions into a complete metric ...
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### Strong ergodicity of a countable subgroup of $PO(3,1)$

If we identify the boundary at infinity of the hyperbolic $3$-space $\mathbb{H}^3$ with the complex projective line $\mathbb{P}^1(\mathbb{C})=\mathbb{C} \cup \{ \infty\}$, we know that the ideal ...
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### Trying to understand why this local coordinates parametrizes a manifold

First of all, I would like to say that I think this question fits better on Math Overflow than on Math Stack Exchange, in view of the proposal of the two sites. However, if my analysis of the ...
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### Maximal ergodic theorem on some dyadic intervals

What we refer to maximal ergodic theorem in this thread is the following: let $\left(\Omega,\mathcal F,\mu\right)$ be a probability space and let $T\colon\Omega\to \Omega$ be a measurable and measure ...
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### Density of closed orbits on hyperbolic surfaces

It is well-known that the set of closed geodesics on a closed hyperbolic surface is dense. My questions: If this property still holds on finite-area hyperbolic surfaces, infinite-area hyperbolic ...
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### Question on a proof of density of periodic orbits

In page 215 and 216 of the book "Introduction to the Modern Theory of Dynamical Systems" by Anatole Katok, Boris Hasselblatt, there is a theorem stated as following: Theorem: Let $\Gamma$ be a ...
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### Generalized right Perron-Frobenius eigenvector with rationally independent coordinates

Suppose you are given a directed graph $G=(V,E)$ which is strongly connected, i.e. for every two vertices $u,v \in V$ there exists a directed path between them. Consider the corresponding edge shift ...
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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 ...
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### Is there a connection $\nabla$ for which this particular non geodesible vector field $X$ satisfy $\nabla_X X=0$?

Let $X$ be the following vector field on $\mathbb{R}^2\setminus \{0\}$ \begin{align} x' &= x\,(1-x^2-y^2)(x^2+y^2-3) - y\,(2-x^2-y^2)\\ y' &= y\,(1-x^2-y^2)(x^2+y^2-3) + x\,(2-x^2-y^2). \...
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### Symplectic forms and sign of eigenvalues

This question has come out while reading J. Moser "New Aspects in the Theory of Stability of Hamiltonian Systems". I'm particularly interested to the Appendix, where one investigates the stability of ...
Inspired by the Lie algebra discussed in this answer, we consider the following Lie agebra associated to a given foliation: Let $\mathcal{F}$ be a nontrivial foliation of a ...