# Tagged Questions

**3**

votes

**2**answers

78 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
$\{...

**-1**

votes

**0**answers

94 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 ...

**0**

votes

**0**answers

59 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**

votes

**0**answers

79 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 ...

**3**

votes

**1**answer

54 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 ...

**5**

votes

**2**answers

84 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 $(X,\mathcal{B},\mu)$ ...

**1**

vote

**0**answers

17 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). ...

**6**

votes

**0**answers

403 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 ...

**-1**

votes

**0**answers

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 ...

**0**

votes

**0**answers

56 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:
\...

**7**

votes

**1**answer

367 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

**0**answers

74 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) ...

**3**

votes

**1**answer

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 ...

**2**

votes

**0**answers

121 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

**0**answers

186 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.
...

**2**

votes

**1**answer

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, ...

**2**

votes

**0**answers

83 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 ...

**1**

vote

**0**answers

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 ...

**5**

votes

**0**answers

62 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 $M$ ...

**8**

votes

**2**answers

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 ...

**1**

vote

**0**answers

108 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 ...

**2**

votes

**1**answer

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 ...

**1**

vote

**0**answers

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 ...

**6**

votes

**1**answer

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

**2**answers

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**

vote

**0**answers

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 ...

**0**

votes

**1**answer

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 ...

**7**

votes

**3**answers

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.)...

**-4**

votes

**1**answer

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{...

**4**

votes

**1**answer

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 ...

**0**

votes

**0**answers

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**

vote

**1**answer

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=-...

**1**

vote

**0**answers

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.
...

**38**

votes

**2**answers

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

**0**answers

92 views

### Connection between cardiac equations and untangling knots?

I was surprised to learn that there is (conjecturally) a connection between a cardiac muscle model known as the FitzHugh-Nagumo equations, and untangling knots:
Maucher, Fabian, and Paul Sutcliffe....

**0**

votes

**0**answers

24 views

### Equivalence classes on an ordered Bratteli diagram

Let $S$ be the adic transformation preserving a probability measure $\mu$ on the set $\Gamma$ of infinite paths of a $\mathbb{N}$-graded ordered Bratteli graph.
For every $n \geq 0$ define the ...

**2**

votes

**0**answers

73 views

### Hausdorff dimension = entropy/Lyapunov exponent for the baker's map?

Let $\Sigma=\{0,1\}^{\mathbb Z}$ and let $\sigma:\Sigma\to\Sigma$ be the left shift. Then it is well known that $(\Sigma, \sigma)$ is conjugate to the baker's map $B$ of the unit square:
$$
B(x,y) = \...

**2**

votes

**0**answers

44 views

### Lyapunov exponents of Lorenz63 and Lorenz96 system

Can someone suggest a reference on the mathematical results (NOT numerical) on the Lyapunov exponents of Lorenz-63 and Lorenz-96 systems (or any other non-trivial system)? In particular, is it always ...

**1**

vote

**1**answer

194 views

### Applications of topology to discrete dynamical systems?

I'd like to know some of the applications of topology to discrete dynamics. By discrete dynamics I loosely mean studying maps between discrete sets.
I mean cases where adding a topology to the sets ...

**0**

votes

**0**answers

48 views

### parametrizations for sections of time in a flow

Let $\phi:\mathbb{R}\times X\rightarrow X$ be a flow in $X$, where $(X,d)$ is a compact metric space. Denote by $\mathcal{H}$ the set of continuous maps $ h:\mathbb{R}\rightarrow \mathbb{R}$ such that ...

**0**

votes

**0**answers

27 views

### A problem from Sakai's book on derivations on C(K) and differential structure on K

In his book, Operator Algebras in Dynamical Systems, at page 59 Sakai poses the following question.
Problem: Let K be a compact space and suppose that C(K) has a non-zero closed *-derivation. Then ...

**1**

vote

**0**answers

76 views

### Is there an area preserving toral endomorphism with critical point?

An endomorphism is a continuous map $f:\mathbb{T}^2 \to \mathbb{T}^2$. An conservative endomorphism is an endomorphism that is area preserving $(m(f^{-1}(U)=m(U), \forall U$ borel set and m is the ...

**5**

votes

**1**answer

68 views

### Survivor sets for expanding maps of the interval

Let $T:[0,1]\to [0,1]$ be a piecewise smooth expanding map, i.e., $|T'(x)|>1$ for all $x$. Let $I_n$ be a sequence of nested intervals (i.e., $I_{n+1}\subset I_n$) such that the length of $I_n$ ...

**2**

votes

**0**answers

27 views

### A proper class for smooth chaotic function

This might be a little, soft, but I'll try
Consider the interval $I=[-1,1]$. We will define a chaotic function $f:\mathbb{R}_+ \times I \to \mathbb{C}$ in the following traditional way:
For every $...

**0**

votes

**1**answer

55 views

### Long term behavior of a certain discrete time dynamical system on graphs

Consider the graph $(V,E)$ with vertex set $V=\{v_1,...,v_n\}$ and edge set $E\subset V\times V$. Further, assume that $\forall v_i\in V, (v_i,v_i)\in E$.
Assume that each vertex has an $\textit{...

**2**

votes

**0**answers

257 views

### Measure of the Attractor of Critical Points of a Manifold

Let $f: \mathbb{R}^d \rightarrow \mathbb{R}$ be a smooth function and consider the $d$-manifold $M = \{(x, f(x)): x \in \mathbb{R}^d\} \subset \mathbb{R}^{d+1}$. Let $P$ be a property of some subset ...

**1**

vote

**0**answers

86 views

### Ergodic skew product on $\mathbb T^d\times U(2)$

Let $\mathbb T^d=\mathbb R^d/\mathbb Z^d$ be the $d$-dimensional torus with normalized Haar measure $\mu_1$ and let $U(2)$ be the group of $2\times2$ unitary matrices with normalized Haar measure $\...

**2**

votes

**0**answers

238 views

### Is there a Bayesian theory of deterministic signal? Prequel and motivation for my previous question

This is a prequel to my question:
What's the probability distribution of a deterministic signal? (functional integrals in probability theory)
Clearly my question looks at the same time fairly ...

**10**

votes

**1**answer

255 views

### About positive upper density

For $S\subset \mathbb{N}$ define the upper density as $D^{\ast
}(S)=\limsup_{n\rightarrow \infty }\frac{\left\vert S\cap \{1,2,\ldots,n\} \right\vert }{%
\left\vert n\right\vert }.$
Question: ...

**1**

vote

**0**answers

30 views

### About stable manifold of a point [closed]

Let $(X, d)$ be a compact metric space and $f:X\rightarrow X$ be a homeomorphism and
$$W^{s}(x)=\{y| d(f^{n}(x), f^{n}(y))\rightarrow as \ n\rightarrow \infty\}.$$
Question: What condition on $(X, ...