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

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11 views

### About the nature of the limiting map $f(x,y).$ obtained via uniform convergence

Let us consider a continuous family of piecewise linear 2-D discrete maps $(f_{n}(x,y))_{n\in\mathbb N}$. Assuming that there exist $m>1$ such that for all $n>m$, each map $f_{n}(x,y)$ is ...

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58 views

### size of local strong stable manifold is measurable

Let $M$ be compact manifold. suppose $f:M\rightarrow M$ is $C^{2}$.
There is a continuous splitting of the tangent bundle $TM=E^{ss}+E^{s}+E^{u}$ invariant under the derivative $Df$ of the ...

**6**

votes

**1**answer

329 views

### Given any finite relation $R$ what is the cardinality of $\langle R\rangle=\{\underbrace{R\circ R\cdots \circ R}_{n\text{ times}}:n\in\mathbb{N}\}$?

Given any finite relation $R$ if we let $\circ$ denote relation composition and define $R^n=\underbrace{R\circ R\cdots \circ R}_{n\text{ times}}$ then does there exist an explicit formula for the ...

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71 views

### Supremum over all invariant Borel probability measures of the ergodic averages ratio of rates

Let $M$ a two-dimensional compact manifold and $f:M\to M$ a diffeomorphism $C^r$, $r\geq 2$ and $f(x,y)=(mx,\lambda y)$ where $m:M\to \mathbb{R}$ and $\lambda:M\to \mathbb{R}$ ,$\lambda<1<m$.
...

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57 views

### Is the Mandelbrot set weakly self-similar?

A subset $F$ of an Euclidean space $E$ will be called weakly self-similar if for all $x \in F$ there is $\epsilon_x>0$ such that for all positive $\epsilon \le \epsilon_x$ there are $y \in F$, $\...

**1**

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**1**answer

48 views

### Lotka Volterra existence of Caratheodory solution

I strive to prove that the following system of differential equations:
$$\begin{cases} x'=x-u(t)xy\\ y'= -y+u(t)xy \\ x(0)=x_0>0\\ y(0)=y_0>0 \end{cases}$$
has a unique Caratheodory solution ...

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48 views

### About the limit of transverse intersection

Let $n$ be a fixed positive integer, and let $W^{s}(R_{q})$ and $W^{u}(R_{q})$ be the stable and unstable manifolds of a fixed point $R_{q}$ of a discrete 2-D mapping. Notice that the sequence $R_{q}$ ...

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votes

**2**answers

209 views

### Newton method and Siegel disks

I am looking for a degree 3 polynomial $P$ whose associated Newton's method $z \mapsto z - P(z)/P'(z)$ has a Siegel disk. Is there an explicit example of such polynomial $P$?

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votes

**3**answers

194 views

### Random reflections unexpectedly produce banded distributions

Start with $p_1$ a random point on the origin-centered unit circle $C$.
At step $i$, select a random point $q_i$ on $C$, and a random mirror line
$M_i$ through $q_i$, and reflect $p_i$ in $M_i$ to ...

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**1**answer

152 views

### Lipschitz property of holonomies fails when stable leaves $W^s(x)$ inside the leaves $W^{ss}(x)$

Let $M$ be compact manifold. suppose $f:M\rightarrow M$ is $C^{2}$.
There is a continuous splitting of the tangent bundle $TM=E^{ss}+E^{s}+E^{u}$ invariant under the derivative $Df$ of the ...

**2**

votes

**1**answer

127 views

### Irreducible but not completely irreducible

Consider a product of i.i.d. $d\times d$ random matrices $A_{i}$
(with $\mathbb{E}\log\left\Vert A_{i}\right\Vert <\infty$).
Let $F:M\times \mathbb R^d\to M\times \mathbb R^d$ be a linear cocycle, ...

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**1**answer

135 views

### continuity entropy with respect gibbs measures

Let $X=\{0,1\}^{\mathbb{N}}$. For simplicity I consider Bernoulli measures on $X$ only.
Let $f:X\to \mathbb{R}$ be Holder continuous. The measure $\mu$ is a Gibbs measure with potential $f$ if there ...

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vote

**1**answer

28 views

### Marginal stability of discrete linear time-invariant system

I have a question about marginal stability of a system:
\begin{equation}
\mathbf{x}[k] = \mathbf{A}\mathbf{x}[k-1]
\end{equation}
I would adapt the definition of marginal stability from this ...

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63 views

### Stable commutator lengths of pseudo-Anosovs

Does anyone have an example of a pseudo-Anosov mapping class for which the stable commutator length is known exactly?

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**1**answer

73 views

### stationary measure for linear cocycle(random transformation matrices)

Let $(M,\mathcal B, \mu)$ be a probability space which $M=\{A_{1},A_{2},...,A_{N}\}^{\mathbb{N}}$ ($A_{i} \in GL(d ,\mathbb{R})$) and $\mu=p^{\mathbb{N}}$. Let $F:M\times \mathbb R^d\to M\times \...

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108 views

### How was the pair of pants introduced [closed]

There are many results mentioned pairs of pants, and it seems to be a classical model. Why are the pairs of pants so useful?
For example, does it have any application if we estimate the perimeter or ...

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**2**answers

289 views

### Minimal, uniquely ergodic but not Lebesgue-ergodic?

So here's my question:
Does there exist a minimal diffeomorphism of class at least $\mathcal{C^2}$ of a compact manifold X which is
minimal
uniquely ergodic with unique probability measure $\mu$
...

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**2**answers

376 views

### Geodesics on hyperbolic surfaces whose closures have arbitrary Hausdorff dimension

Consider the geodesic flow on $X = \Gamma \backslash \text{PSL}(2,\mathbf{R})$, the unit tangent bundle of a hyperbolic surface, where $\Gamma$ is a lattice.
I have heard that, for any real number $\...

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29 views

### definition of mixing component

definition of ergodic component: consider stationary dynamical system $(X, \mathcal{B}, \mu, T)$, each ergodic component is $m(\cdot)=\mathbb{E}_{\mu}^{\mathcal{I}}\mathbf{1}_{(\cdot)}$, $\mathcal{I} $...

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86 views

### Is $\partial M_d$ continuously determined by $d$?

This question is inspired by a question on math.stackexchange:
https://math.stackexchange.com/questions/1707291/is-the-generalized-mandelbrot-set-a-fractal-in-the-d-dimension/2575089
The animation ...

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90 views

### How many two-dimensional space filling Hilbert-like curves are there?

I'm interested in filling 2d square with space filling, non-self-intersecting, locality preserving, self-similar curves, like Hilbert curve. I found interesting work concerning three dimensional case ...

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**1**answer

57 views

### Does differentiating an integro-differential equation results in equivalent stability of the solution?

I have a dynamical system in the form of an integro-differential equation which I want to analyze in terms of stability. To demonstrate my problem consider the following integro-differential equation:
...

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55 views

### 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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104 views

### 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 }...

**3**

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**0**answers

212 views

### 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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**3**answers

625 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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**2**answers

106 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,...

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104 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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201 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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56 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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**1**answer

344 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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**1**answer

67 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 ...

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28 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 ...

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**0**answers

80 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 ...

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40 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 ...

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**1**answer

129 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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**1**answer

94 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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18 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 $...

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**0**answers

65 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 ...

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**1**answer

107 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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94 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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73 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 ...

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**1**answer

187 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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**0**answers

86 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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**1**answer

180 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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**1**answer

118 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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52 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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**0**answers

72 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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30 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 ...