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

7
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2answers
202 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 ...
4
votes
2answers
88 views

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,...
0
votes
0answers
94 views

Integral of “Torsion” and “Curvature” as two functionals in variational approach to the Hilbert 16th problem

Let $X$ be a polynomial vector field on the real plane and $\gamma$ be a closed orbit of $X$. Then $$\int_{\gamma} \kappa(s)ds=\pm 2\pi \qquad (1)$$ where $\kappa$ is the curvature of the curve $\...
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votes
0answers
42 views

Infinite singular point of algebraic curve [on hold]

I have an algebraic curve $-(3/2) x^2 y^2 + (1/4) y^4 - c/2 x^2 = 0$ with $c<0$. This curve has $(1,0,0)$ as an infinite singular point. Please help me to know its nature.
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0answers
138 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 ...
2
votes
0answers
41 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 \...
3
votes
1answer
322 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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1answer
48 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 function in a complete metric ...
1
vote
0answers
26 views

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 ...
3
votes
0answers
72 views

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 ...
1
vote
0answers
31 views

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 ...
3
votes
1answer
123 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(...
1
vote
0answers
41 views

small perturbation of transfer operator without discrete spetrum

Pommeville-Manneau maps: $ T_{\alpha}=x+2^{\alpha}x^{\alpha+1} x \in [0,\frac{1}{2}], 2x-1, x \in [\frac{1}{2}, 1], \alpha <1$ is well known to have polynomial decay of correlation, it transfer ...
2
votes
1answer
90 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}, ...
0
votes
0answers
15 views

Are there any results on “degenerate” slow manifolds where the fast system yields a conserved quantity?

I assume that readers of this question is familiar with general results in geometric perturbation theory and fast-slow systems of the form $\dot{x} = f(x,y),\quad \epsilon \dot{y} = g(x,y)$ with $...
2
votes
0answers
59 views

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 ...
6
votes
1answer
104 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 ...
4
votes
0answers
55 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\...
3
votes
0answers
88 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 ...
3
votes
0answers
71 views

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 ...
3
votes
1answer
180 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 ...
2
votes
0answers
63 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 ...
9
votes
1answer
175 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 ...
7
votes
1answer
113 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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votes
0answers
43 views

Stability of an equilibrium for a third order control system with an integral regulator limited by the stop-type element

In what publication is the following theorem prove carried out? The method of the Lyapunov functions is preferable. Thanks. Consider the system consisting of the controlled object and regulator. The ...
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vote
0answers
66 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 ...
0
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0answers
29 views

Reference request for study of fractals with non invariant similarity scaling

I have heard recently that there is some study but not much progress on the dynamics of random walks (or fractals?) with non-invariant scaling. This is in contrast to brownian motion and conventional ...
5
votes
0answers
100 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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0answers
105 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"...
2
votes
1answer
232 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 ...
2
votes
1answer
162 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 $...
0
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0answers
92 views

norm and conorm of elliptic cocycle be different

Let $(M,\mathcal{B},\mu)$ be a probability space and $f:M \rightarrow M$ be a measure preserving map.Let $A:M \rightarrow SL(2,\mathcal{R})$be a measurable function with value invertiable $2\times2$...
1
vote
0answers
39 views

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 ...
1
vote
0answers
61 views

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 ...
2
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0answers
86 views

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 ...
1
vote
3answers
203 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 ...
8
votes
0answers
351 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 $\omega_{1}$...
6
votes
1answer
517 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(\...
3
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0answers
123 views

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,\...
4
votes
2answers
218 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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0answers
68 views

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 ...
0
votes
1answer
79 views

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 ...
2
votes
0answers
72 views

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 ...
6
votes
0answers
107 views

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\...
3
votes
1answer
161 views

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). \...
4
votes
1answer
188 views

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 ...
1
vote
1answer
160 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 ...
5
votes
1answer
223 views

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 $...
3
votes
2answers
765 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 ...
2
votes
0answers
58 views

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, ...