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

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4
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2answers
100 views

A question on invariant measures

Let $(X, \mathcal{B}, T)$ be a topological dynamical system and $M(X, T)$ be the set of all invariant measures. I do not know is there some nice functional characterization of the following set $\{...
0
votes
1answer
65 views

Convergence of Discrete Geodesic

Let $M$ be a Riemmanian manifold, $p\in M$ and $V\in T_p(M)$. Suppose $f^{-1}:U_p \mapsto U$ is a diffeomorphism of a neighborhood of p to an open subset of $\mathbb{R}$ and define the sequence: \...
0
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0answers
25 views

A certain measure on $C^{*}$algebras

Is there a reference who introduce the following measure on a $C^{*}$ algebra and investigate it from the view point of $C^{*}$ algebra or from the view point of ergodic theory?: Let $A$ be a $C^{*}$ ...
3
votes
1answer
57 views

When is a limit cycle generated by a Hamiltonian oval stable?

Consider a real polynomial $H$ of degree $n+1$ in the plane. A closed, connected component of a level curve $H=t$ is denoted by $\gamma(t)$ and called an oval of $H$. Let $\omega$ be a real 1-form ...
19
votes
1answer
704 views

Time averages and differentiability

Let $\varphi_t : M \rightarrow M$ be a smooth flow on a smooth manifold $M$. We may assume (although I'm not sure if this is important) that the flow preserves a smooth volume form on $M$. Given a ...
-1
votes
0answers
96 views

(Dynamical) mean field theory for mathematicians? [on hold]

I am looking for a readable introduction/tutorial on dynamical mean-field theory, written for someone who doesn't know anything about particle physics. My physics background is non-existent beyond ...
6
votes
0answers
409 views
+100

Counting limit cycles via curvature in Riemannian geometry

In this post we would like to give a possible new approach to the second part of the Hilbert 16th problem First we give a short introduction: A quadratic system is a polynomial vector field on ...
0
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0answers
60 views

Application(s) of complex dynamical system into some other areas of mathematics [on hold]

Complex dynamical system is a very active branch in mathematics. I wondered are there some nice applications of complex dynamical system into the some other areas of mathematics.
-1
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0answers
80 views

Are the theorems of Ergodic Theory valid for non-probability spaces?

The theorems in Ergodic Theory have assumed a probability measure, always. I am interested to know if they hold even when the space is not equipped with a probability measure. In other words, if my ...
5
votes
2answers
85 views

Normalization in Birkhoff's theorem and its failure in infinite measure spaces

I have two questions, somehow related. The first one, has to do with the Birkhoff's ergodic theorem. In its classical formulation, it states that if we have a probability space $(X,\mathcal{B},\mu)$ ...
90
votes
12answers
18k views

What is an integrable system

What is an integrable system, and what is the significance of such systems? (Maybe it is easier to explain what a non-integrable system is.) In particular, is there a dichotomy between "integrable" ...
16
votes
1answer
572 views

Example of a quasi-Bernoulli measure which is not Gibbs?

Let $X=\{0,1\}^{\mathbb{N}}$. For simplicity I consider measures on $X$ only. A measure $\mu$ is quasi-Bernoulli if there is a constant $C\ge 1$ such that for any finite sequences $i,j$, $$ C^{-1} \...
1
vote
0answers
18 views

Generalizing an expected increase in autocorrelation near a bifurcation point to a system of ODE

Near a bifurcation point, a stochastically forced dynamical system should show an increase in autocorrelation and variance. This is due to critical slowing (a loss in resilience to perturbations). ...
27
votes
2answers
5k views

The error in Petrovski and Landis' proof of the 16th Hilbert problem

What was the main error in the proof of the second part of the 16th Hilbert problem by Petrovski and Landis? Please see this related post and also the following post. Added : According to their ...
4
votes
2answers
1k views

Invariance of dynamical system under a transformation

I have come across an interesting property of a dynamical system, being transformed by a map, but i haven't been able to figure out why this is happening (for quite some time now actually). Any help ...
-1
votes
0answers
24 views

Complex fixed points on the bifurcation diagrams [closed]

I'm working with bifurcation diagrams, an extesion that is being made of them is the determination of complex fixed points in addition to the real fixed points. Given a dynamic system (e.g. an ...
12
votes
1answer
486 views

resampling over Bowen balls

Hello MO World I'm working on a paper involving embedding your favourite measure-preserving transformation into a topological model (think Krieger generator theorem: embedding in a full shift) and ...
3
votes
1answer
129 views

Stability of a linear system and spectrum of the product of two matrices

Let us consider an invertible matrix $\mathbf{A}\in GL_d(\mathbb{R})$ such that all its diagonal entries $\mathbf{A}_{ii}=-1 \; \forall \, i$. My question is the following: Does it always exists a ...
3
votes
3answers
224 views

Maximizing entropy under constraints

This question is about an extension of the variational principle in thermodynamical formalism when one adds linear constraints to the measures. Consider the one-sided shift $\sigma:\mathcal{A}^\...
0
votes
1answer
70 views

Sequence of translation surfaces and length of saddle connections

A sequence of translation surfaces $(X_n,\omega_n)$ is said to "go to infinity" if it leaves every compact set in the space of translation surfaces as $n$ goes to infinity. I know that this is ...
2
votes
0answers
84 views

If $f$ is dynamically coherent, is there a unique invariant foliation tangent to $E^{c}$?

Let $f$ be a diffeomorphism of a closed manifold $M$ such that $f$ is partially hyperbolic if the tanget bundle of $M$, $TM$ splits into three invariant sub-bunbles $$ TM = E^{s} \oplus E^{c} \oplus ...
3
votes
1answer
79 views

On the $\omega$-limit set of a trajectory converging to a submanifold

Let $X$ be a $C^1$ vector field on $\mathbb{R}^n$. Let $S$ be a compact submanifold of dimension $s(<n)$. Suppose $S$ is invariant under the flow of $X$ and that we know everything about the ...
7
votes
1answer
368 views

Does 53 diverge to infinity in this Collatz-like sequence?

This function has been explored a bit at MSE (over the past week): \begin{eqnarray} f(n) &=& (n-1)^2 \; \textrm{if} \; (n \bmod 4) = 1\\ f(n) &=& \lfloor n/4 \rfloor \; \textrm{...
0
votes
0answers
76 views

Orbit intersection in toral automorphisms

Let $f:(\mathbb{R/Z})^2\to (\mathbb{R/Z})^2$ be a hyperbolic toral automorphism induced by a matrix $A\in GL_2(\mathbb{Z})$. Consider a measurable (wrt to the Lebesgue measure $\mu$ on the torus) ...
2
votes
0answers
122 views

Gaussian Integrals and Pseudo-Anosov Maps

The hep-th section of arXiv if often filled with beautiful semi-rigorous computations on Mathematics. However sometimes it is very difficult to understand what is being stated. Here I take from: ...
7
votes
0answers
187 views

horocycle flow and the prime number theorem

Looking at Zagier's Eisenstein Series and the Riemann Zeta Function, we get a proof of the prime number theorem using horocycles. I would really love it if there were a geometric proof like this. ...
8
votes
2answers
170 views

Interior periodic points of area preserving homeomorphisms of a pair of pants

A celebrated result of Franks shows that any area preserving homeomorphism of the closed annulus $A$ with at least one periodic point (possibly along the boundary) has infinitely many interior ...
2
votes
1answer
82 views

IFS maps on circle

A systems $<f_0,f_1>$ is minimal if the set $\{h(x): h=f_{i_n}\circ f_{i_{n-1}}\circ...\circ f_{i_1}, i_k \in \{0,1\},n>0\}$ is dense in $S^1$, for every $x\in S^1$. Consider $f:S^1 \to S^1, ...
1
vote
0answers
32 views

Differential inequalities for a strictly diagonal dominant system of linear ODEs

Let $A$ be a real $d\times d$ matrix. The diagonal elements are strictly negative ($a_{ii}<0$) and the off-diagonal elements are non-negative ($a_{ij}\geq 0$ for $i\neq j$). $A$ is strictly column ...
1
vote
0answers
268 views

The “Rolle theorem” for sections of a vector bundle

1) Assume that $E\to M$ is a smooth real vector bundle and $\nabla$ is a connection. (We do not assume any metric compatibility since we do not fix a metric on $E$). Assume that ...
7
votes
0answers
833 views

The integral of torsion

I found the following * exercise (exercise *9) in page 407 of the book of do Carmo "Differential geometry of curves and surfaces". This problem is a classical theorem which is referenced in the book ...
2
votes
1answer
177 views

Is there a matrix that converts the gradient of every possible function to gradient of other function?

I have already asked this question on math.stackexchange.com http://math.stackexchange.com/questions/1789476/is-there-a-matrix-that-converts-the-gradient-of-any-function-to-gradient-of-othe Now I ...
5
votes
0answers
63 views

Uniform approximation of a continuous flow by a $\mathcal{C}^1$ flow

Setup: Consider a (smooth) compact Riemannian manifold $M$, whose distance is denoted by $d$. Let $\Phi$ be a continuous flow, namely a continuous application from $\mathbb{R} \times M $ to $M$ ...
1
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0answers
109 views

What interesting information can be deduced from knowledge of how deep a geodesic ventures into the cusp

First of all I have to apologise as I am not a geometer and my knowledge of geometry is poor. Let $M$ be the modular surface and $\gamma$ to denote a geodesic in $M$. In the the following paper by ...
29
votes
5answers
1k views

Surfaces filled densely by a geodesic

Which smooth, closed surfaces $S \subset \mathbb{R}^3$ have no single geodesic $\gamma$ that fills $S$ densely? Say a geodesic $\gamma$ "fills $S$ densely" if the closure of the set of points ...
1
vote
0answers
66 views

Group actions on principal groupoids

Suppose that $\mathcal{G}$ is etale principal groupoid and that $G$ is a discrete (or finite) group acting freely on the locally compact unit space $\mathcal{G}^0$ (or assuming compactness, if ...
7
votes
2answers
219 views

On the density of the sequence $\{n \{n \xi \} \}_n$

I have a question that I can't manage to answer myself. It comes from some work in PDE theory, but it is related to analytic number theory. Let us say that we have an irrational number $\xi$. The ...
7
votes
3answers
132 views

Ball ricochetting from a plane of close-packed spheres

Suppose the lower $z \le 0$ halfspace of $\mathbb{R}^3$ is filled with a rigid close-packing of unit-radius spheres. (I don't think it matters much for my purposes if it is an FCC or an HCP packing.)...
6
votes
1answer
94 views

Is there a complete Riemannian manifold with infinite volume whose the time-one map of the geosesic flow is recurrent?

Let M be complete Riemannian manifold M with infinite volume, it is know that the geodesic flow, $\varphi^t:T^1M \rightarrow T^1M$ preserves the Liouville measure $\mu$, that is, $\mu(\varphi^t(A)) = ...
2
votes
2answers
82 views

Periodic orbits in the plane

Consider a vector field $F:\mathbb{R}^2\rightarrow \mathbb{R}^2$ of the following form $F(y_1,y_2)=(y_2,\mu(y_1))$, where $\mu\in\mathscr{C}^1(\mathbb{R})$ has appropriate growth so that the solutions ...
1
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0answers
41 views

Hyperbolic PDE from total derivative?

Given a density function $p(t, \boldsymbol{x})$, where $t$ is time and the vector $\boldsymbol{x}$ represents a point in $n$ dimensional space, a hyperbolic PDE describing the time evolution of the ...
38
votes
2answers
2k views

Can we trap light in a polygonal room?

Suppose we have a polygonal path $P$ on the plane resulting from removal of an one of a convex polygon's edges and a ray of light "coming from infinity" (that is, if we were to trace the path ...
-4
votes
1answer
115 views

Existence and uniqueness of solutions for a system of first order PDEs [closed]

Which results can be applied and which conditions are needed, to ensure the existence and uniqueness of the solutions of the first order of PDEs: A$\dfrac{\partial}{\partial t}\pmb{v}(t,x)=B(t,x,\pmb{...
12
votes
2answers
408 views

Nuclear operators/spaces and transfer operators

While studying for my thesis (in dynamical systems) I've encountered multiple times with the concept of nuclear operators and nuclear spaces, often linked with the works of Grothendieck. For example, ...
13
votes
3answers
1k views

Periodic orbits and polynomials

There are two simple and classic enumerations that still I'm puzzled about. Let's start with a simple counting problem from a well-known dynamical system. fact 1 Consider the "tent map" f:[0,1]→[...
4
votes
1answer
101 views

Ergodic, non-atomic measure on the circle which are $\times 2$ and $\times \frac12$ invariant

There any many ergodic, $T$-invariant, non-atomic measures on the space $X = [0,1)$, where $Tx = 2x \pmod 1$ is the doubling map. My question is: are any such measures also $T^{-1}$-invariant? BYO ...
6
votes
0answers
211 views

Topological entropy and periodic sequences of a subshift

Let $\Sigma$ be a two-sided subshift on a finite alphabet $A$. Let $\Sigma_n$ denote all words $x_{-n}\dots x_n\in A^{2n+1}$ such that $(x_k)_{-\infty}^\infty \in \Sigma$ for some $x_k, |k|>n$. ...
1
vote
1answer
119 views

Is there an entire solution for the Van der pol equation?

Is there a non constant entire function $\gamma(t)=(x(t),y(t)): \mathbb{C} \to \mathbb{C}^{2}$ which satisfy the following Vander pol dififferential equation? $$\begin{cases}\dot{x}=y-x^{3}\\\dot y=-...
0
votes
0answers
41 views

Necessity of expansiveness for existence absolutely continuous invariant measures for piecewise smooth maps of an interval

A map $\tau:[0,1]\to[0,1]$ is piecewise smooth (or $C^r$) if there is a partition of $[0,1]$ into intervals, $[0,1]=\cup I_n$, (which can be either finite or countable) such that the restriction of $\...
1
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0answers
58 views

Dynamics of pairwise distances in the $n$-body problem

Disclaimer: I have asked this question on Physics SE a week ago, but got no answers. I know that some MO users are interested in the $n$-body problem, so I decided to cross post here as well. ...