# 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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### 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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### Renyi's theorem on mixing

I have been trying to understand the proof of Renyi's characterization of (strongly) mixing transformations:
A measure preserving transformation $T \text{ is strongly mixing iff for every measurable }...

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### Periodic orbit for certain Hamiltonian on the tangent bundle

In this question a nontrivial periodic orbit is a periodic orbit which is not a singular point.
Let $p: \mathbb{R}^n \to \mathbb{R}$ be a ...

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657 views

### Has dynamics on $G/\Gamma$ ever been used to prove interesting things about $\Gamma$?

Fix a Lie group $G$ and a discrete subgroup $\Gamma \subset G$. Homogeneous dynamics is about studying the actions of subgroups $H \subset G$ on the quotient $G/\Gamma$.
Does anyone know of an ...

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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,...

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215 views

### If two group actions lead to the same orbifold, are they conjugate?

In The Geometry and Topology of Three-Manifolds, Thurston says: "In these examples, it was not hard to construct the quotient space from the group action. In order to go in the opposite direction, we ...

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117 views

### Asymptotic colouring of edges and vertices, and untwisting cocycles

This question regards colourings on edges and vertices on countable directed multigraphs.
We start with an example. Let $G=\mathbb Z^2$. We define two functions $a_h$ and $a_v$ from $\mathbb Z^2$ to $\...

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372 views

### 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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103 views

### 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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### Limit contration rates and expansion rate solenoid map

Let M:=$S^{1}\times \mathcal{D}^1$ where $\mathcal{D}=\{v\in \mathcal{R}^2 | |v|<1\}$ carries the product distance and suppose $f:M\rightarrow M$,$(x,y,z)\rightarrow (\gamma x, \lambda y+v(x), \mu ...

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138 views

### Is the convergence of $\dot{x}=2A(t)x$ faster than that of $\dot{x}=A(t)x$?

Let $x \in \mathbb{R}^{n}$ and $A(t) \in \mathbb{R}^{n\times n}$. If $\dot{x}=A(t)x$ and $\dot{x}=cA(t)x$ with $c>1$ are exponentially stable. Is the convergence rate of $x$ to zero of $\dot{x}=cA(...

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102 views

### time delay ergodic theorem

given dynamic system $(X, \mathcal{B}, F, \mu), \mu \circ F^{-1}=\mu, F $ is mixing, $ A \in \mathcal{B}, s.t. \mu(A) >0 $.
consider dynamic system $(X\times X, \mathcal{B}\otimes \mathcal{B}, ...

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### Stability of dynamical system via Lyapunov

I have a dynamical system which has the following form:
$\dot x=\mathcal F_1(m_1)x+\mathcal F_2(m_2)x$.
My objective is to find the parameters $m_1$ and $m_2$ via LMI (linear matrix inequality) using ...

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118 views

### Existence of a real eigenvalue is a necessary condition for the density of all the orbits of a Lie subgroup of $GL(\mathbb{R},d)$

Good morning,
I would like to pose the following (maybe naive) question. Let $\mathfrak{a}\subset \mathfrak{gl}(\mathbb{R},d)$ be any lie subalgebra, and $A$ be the connected, simply connected ...

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56 views

### A geometric quantity associated to a vector field on a surface

Let $(M, g)$ be a $2$ dimensional Riemannian manifold.
Then we consider the Riemannian metric on TM described here.
Assume that $X:M\to TM$ is a vector field. For every $p\...

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102 views

### 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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197 views

### 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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92 views

### 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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184 views

### 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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120 views

### Special Cases of Duistermaat-Heckman Formula

The Duistermaat Heckman localization formula states how integrals over symplectic spaces with Hamiltonian $U(1)$ group actions.
$$ \int_M \frac{\omega^n}{n!} e^{-\mu} = \sum_{x_i \text{ fixed}} \frac{...

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83 views

### Computing algebraic entropy

Could you recommend any reference for computing algebraic entropy?
Here algebraic entropy is defiened as $\lim_{n \to \infty}\log (deg (f^n))^{1/n}$ for a rational map $f $.
I saw that there are ...

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104 views

### Algebraic independence of limit cycles of Lienard equation

It is well known that the Van der Pol equation $$\begin{cases} \dot x=y-(x^3-x)\\\dot y=-x \end{cases}$$ has no an algebraic limit cycle.
According to this fact, we search for a related ...

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169 views

### Desingularization of the zero section of $TM$ as the manifold of singularities of the geodesic flow

However the method of "Blowing up of singularities" is initially introduced for an isolated singularity, but this method have been generalized to blowing up of a "Manifold of singularities"...

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244 views

### Is the exceptional set Zariski open?

Let $T$ be a smooth projective variety and $f_T : \mathbb {P}^N_T \rightarrow \mathbb {P}^N_T $ be a family of dominant rational maps. The dynamical degree of a dominant rational map $f $ is defined ...

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174 views

### fast algorithms for external angle computations

Two related problems related to the complex quadratic polynomial $f_c(z) = z^2 + c$ and Mandlebrot and/or Julia sets:
find an external angle $\theta_c$ for a complex point $c$
find a complex point $...

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### Center Manifold Theorem and case of all zero eigenvalue

Is the center manifold theorem applicable if say for a planar(2D) system of non-linear ode, the stability matrix has both eigenvalues zero? Of course, there is only one eigenvector.
If not, what is ...

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### Integrability, quantum ergodicity, and observable algebra

Consider (for simplicity and definiteness) the Laplacian on a compact Riemannian manifold $M$. Let $\phi_k$, $E_k$ be its eigenfunctions and eigenvalues in increasing order. Quantum ergodicity is ...

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### coboundary in Dynamical system

a question about the definition:
given measurable dynamic system $ ( X, \mathcal{B}, T, \mu)$, $ \mu \circ T^{-1}=\mu$ ergodic.
$\phi \in L^{\infty}$ is coboundary with $\int \phi d\mu =0 $, means ...

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209 views

### Repeatedly halve and twist a planar shape: Limiting shape?

Consider the following iterative process.
Start with a planar region $R=R_0$ of $\mathbb{R}^2$.
I am thinking of $R$ as connected,
but it may become disconnected.
In the example below, $R$ starts as ...

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364 views

### Determinant as a Hamiltonian

Are there two symplectic structures $\omega_1, \omega_2$ on $M_{2n}(\mathbb{R})$ such that the function $Det:M_{2n}(\mathbb{R})\to \mathbb{R}$ is completely integrable with respect to $\omega_{1}$...

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622 views

### (In)stability of a two-dimensional dynamical system

Consider the following system of coupled differential equations
$$
\dot{x}_1(t) = -x_1(t) - \cos(\omega t)x_1(t) + \cos(\omega t)x_2(t), \ x_1(0)\in\mathbb{R},\\
\dot{x}_2(t) = -\gamma x_2(t) - \cos(\...

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### Random $\beta$-transformation and its limit theorem

given probability space $ (\Omega, T, \mu), \mu$ is ergodic and $ T $ is invertible ( can regard $T$ as two sides shift)
define random $\beta$-transformations: random variable $\beta:\Omega \to (1,\...

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239 views

### A kind of converse to the Hopf theorem on ergodicity of geodesic flow in negative curvature

Is there a 2 dimensional Riemannian manifold $M$ whose curvature is not negative but its geodesic flow is an ergodic flow?

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### On the measure of regular and chaotic regions in a phase space

Consider a Hamiltonian, non-linear, dynamical system associated to $H(\vec{q},\vec{p})$. Assume that the number of effective degrees of freedom is relatively small, say $D=3,4,5$. Now choose a certain ...

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### Ergodicity of geodesic flow in negative curvatutre as a possible obstruction for consideration of limit cycles as closed geodesics(4)

Does the ergodicity of geodesic flow of compact surfaces with negative curvature stile hold for non compact case?
Is not the ergocity theorems of geodesic flow an obstruction to have a ...

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### Counting orbits of the standard map

Consider the standard map. Might it happen that for some nonzero parameter value $K$ and for some positive integer $q$ that there exist an infinite number of periodic orbits having period $q $ I ...

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### Conditions to the existence of periodic orbits of non vanishing vector fields on $\mathbb{T}^2$

I'm doing a research about Filippov systems on $\mathbb{S}^3$ with discontinuities on $\displaystyle\frac{1}{\sqrt{2}}\cdot\mathbb{T^2} =\left\{\displaystyle\frac{1}{\sqrt{2}}x ; \ x \in \mathbb{S}^1\...

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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 ...

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166 views

### A Lie algebra associated to a foliation(A kind of saturation of foliations)

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 ...

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### Teichmueller disk and the $\mathrm{SL}_2\mathbb{R}$ action

Let $(X,\omega)$ be a Riemann surface of genus $g$ with holomorphic 1-form $\omega$ (or equivalently a translation structure). Let $\Omega\mathcal{T}_g$ be the space of holomorphic 1-forms over genus $...

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808 views

### On Mathematical Foundations of Football

Football (soccer) is arguably one of the most unpredictable sports. Countless variables play a role in determining the outcome of a certain football match. Due to the high complexity of the entire set ...

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### Does there exist a leaf of this holomorphic foliation with non trivial holonomy?

Let's $\mathcal{F}$ be the holomorphic foliation of $\mathbb{C}^2$ tangent to the kernel of $\alpha=(sin x) dx -(cos x)dy$.
Are all leaves of $\mathcal{F}$ simply connected? If the answer is no, ...

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### Automorphisms of Erdös spaces

It is well-known that there is an automorphism of the Cantor set $h:C\to C$ such that $\overline{\{h^n(c):n\in \mathbb Z\}}=C$ for every $c\in X$.
In other words, there is a self-homeomorphism of $C$...

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### Has this type of pathwise (S)DE been studied before?

I thought of a possible type of pathwise-defined nonautonomous/stochastic differential equation, and I was wondering if it has been studied before.
Let $(G,\ast)$ be an abelian $C^1$ Lie group....

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### Two questions on “foliation by geodesics”

I would appreciate if you consider the following two questions on $1$ dimensional foliations whose leaves are geodesic.
1)Assume that $M$ is a Riemannian manifold which is either an open ...

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178 views

### $\{x^n\}$ dense in $[0,1]$ for rational noninteger $x > 1$ [closed]

Is the sequence $\{\{x^0\}, \{x^1\}, \{x^2\}, \cdots \}_{n \geq 0}$ dense in $[0,1]$ for rational noninteger $x > 1$, where $\{x\}$ denotes the fractional part of $x$, i.e $\{ x \} = x - \lfloor x \...