**1**

vote

**0**answers

107 views

### Size of the eigenfunction of Laplacian (reference request)

It is a classical Sobolev inequality that if $\phi$ is an eigenfunction of the Laplace-Beltrami operator on a $n$-dim compact Riemannian manifold $M$ with eigenvalue $\lambda$ then
...

**8**

votes

**3**answers

208 views

### Uniquely ergodicity and polynomial ergodic average

Let $(X,T)$ be a uniquely ergodic system (here X is compact, T is a continuous map form $X$ to itself), so for any continuous function $f:X\rightarrow\mathbb{R}$ we have for any $x\in X$, the ergodic ...

**5**

votes

**2**answers

243 views

### Examples of surface automorphisms with no periodic points

Consider a smooth projective complex surface $S$ with an automorphism $g:S\to S$. A point $p$ is periodic if it has finite orbit under iterates of $g$.
What are some examples of surface ...

**0**

votes

**0**answers

93 views

### Invariant mesures for expanding maps of the circle

Is there any characterization for the support of T-invariant measures? where T is a C¹ expanding map of the circle i.e. T'(x)>Lambda>1 for all x in the circle.
I know there are periodic and total ...

**3**

votes

**2**answers

154 views

### Differential form equation in Bowen's lecture notes

I'm reading one of the classical theorems presented in Bowen's lecture notes, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms." I'm trying to figure out a very short line of ...

**9**

votes

**1**answer

134 views

### Nonconventional ergodic averages for commuting transformations

Let $S$ and $T$ be commuting measure-preserving transformations of a standard probability space $(X,\mu)$, so $S$ and $T$ define an action of $\mathbb{Z}^2$ on $(X,\mu)$. I am wondering about ...

**11**

votes

**2**answers

3k views

### *The* open problem in General Relativity?

Q. Is there a single, clear mathematical question that has emerged as
the open problem in General Relativity?
I ask this on the ~100th anniversary of Einstein's (4-page!) 1915 paper,
"Die ...

**2**

votes

**1**answer

191 views

### Embeddings of subshifts

Consider $(X,\sigma_X)$ and $(Y, \sigma_{Y})$ be subshifts of the one sided shift in two symbols. Assume that $(X,\sigma_X)$ is a transitive subshift of finite type and $(Y, \sigma_{Y})$ is a ...

**2**

votes

**1**answer

176 views

### Reading Ratner's paper “Ragunathan's conjectures for SL(2,R)”

Hello everyone (interested),
I am trying to read Marina Ratner's paper "Ragunathan's conjectures for $SL_{2}(R)$" (Israel Journal of Mathematics 80 (1992), 1-31). There is a claim right at the end of ...

**3**

votes

**0**answers

103 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 ...

**5**

votes

**0**answers

111 views

### Hyperbolic toral automorphisms, and maximizing over orbits the minimum along an orbit of a function

Setup:
Let $\phi\colon T^2 \to T^2$ be a hyperbolic toral automorphism. Let $f\colon T^2 \to \mathbb{R}$ be a continuous function.
For $x \in T^2$, let $\underline{f}(x) = \inf_{n \in \mathbb{Z}} ...

**0**

votes

**0**answers

33 views

### On the stability analysis of a discrete difference system with multiplicative noise

If we assume that
\begin{equation*}
\rho \{\phi \otimes \phi+\psi \otimes \psi\}<1,
\end{equation*}
where $\rho$ denotes the spectral radius, then can we verify that the following inequality ...

**9**

votes

**1**answer

471 views

### A question about Mirzakhani et. al.'s algebraicity theorem

While the geodesic flow on a complete hyperbolic surface is ergodic, the closure of an individual orbit (a geodesic line) can take a complicated fractal-like shape. Nonetheless, there is an ...

**10**

votes

**4**answers

591 views

### Rounding errors in images of Julia sets

One typically computes Julia sets by iterating a complex function, such as a polynomial or rational function.
How do rounding errors affect the results?
I'm looking for references on this issue, ...

**11**

votes

**0**answers

322 views

### Does Langton's ant cover every n by 6 gridded torus?

This post follows this other post about times cover by Langton's ant of $n$ by $n$ gridded torus.
For $n$ by $n$ gridded torus, I've checked for $n \le 1000$ that the ant covers all. This fact needs ...

**1**

vote

**2**answers

157 views

### Density of periodic points and density of periodic measures

There are many results (usually connected to specification-like properties) about density of periodic measures in the space of all invariant ones. However some questions that seem to be easy (at first ...

**1**

vote

**1**answer

104 views

### Does every measure-preserving dynamical system admit a backward orbit?

This seems like a really basic question, and yet I haven't managed to find the answer!
Let $(X,\Sigma,\mu,T)$ be a measure-preserving dynamical system. Does there necessarily exist at least one ...

**1**

vote

**2**answers

103 views

### Lyapunov exponents of dual / adjoint / transpose random dynamical system (RDS)

Consider the the state of a system at time $n$, $X_n$, as the action of a product of i.i.d. $d\times d$ random matrices acting on a $d$ dimensional vector $X_0$, so we have
$$X_n = A_n \cdots ...

**19**

votes

**0**answers

526 views

### Time for Langton's ant to cover a “square” torus

Langton's ant is a cellular automaton running as follows:
Squares on a plane are colored variously either black or white. We
arbitrarily identify one square as the "ant". The ant can travel ...

**2**

votes

**1**answer

166 views

### length comparison on negatively curved surfaces

Suppose $g_1$, and $g_2$ are two Riemannian metrics on a closed surface $S$, provided that the Gaussian curvature $K_{g_1}$ $<$ $K_{g_2}\leq -1$. Denote by $\mathcal{C}$ the set of free homotopy ...

**14**

votes

**1**answer

788 views

### In how many steps a random walk visits all the elements of a finite group, with a probability 1/2?

This question is a variation of the return to the origin problem.
Let $G$ be the finite group $\mathbb{Z}/n \times \mathbb{Z}/n$ and let the random transformation $T: G \to G$ such that $T(a,b) = ...

**2**

votes

**0**answers

174 views

### Discontinuity of Radon-Nikodym derivative for Patterson-Sullivan measures for word metrics on Gromov hyperbolic groups

Let $\Gamma$ be a Gromov hyperbolic group coming endowed with a word metric coming from some finite generating set. Let $\nu$ be an associated Patterson-Sullivan measure (quasi-conformal density).
I ...

**0**

votes

**0**answers

110 views

### Product of two foliations

1.What is an example of a manifold $M$ with two foliations $F$ and $F'$ which are not topological equivalent but the product foliations $F\times F$ and $F'\times F'$, as foliations on $M\times M$, ...

**5**

votes

**1**answer

152 views

### Classification of ergodic measures for circle expanding maps

Let us consider the classical self-covering of the circle $S^1=\mathbb{R}/\mathbb{Z}$ given by
$$\times_d(x) = dx \mod 1$$
where the degree $d$ is any integer greater than $1$.
There are a wealth of ...

**1**

vote

**0**answers

43 views

### Techniques for finding the stationary state of a continuous-state, discrete-time Markov process

I'm interested in a continuous-state, discrete-time Markov process. Let the distribution at time $t$ be $f_t(x)$. The update equation has the form
\begin{equation}
f_{t+1}(x) = \int f_t(x') g(x', x) ...

**8**

votes

**0**answers

172 views

### Ricocheting pinball-like shot: Complexity?

Suppose one has $n$ perfect two-sided mirror segments in the plane $\mathbb{R}^2$.
The segments are open, excluding their endpoints.
They are disjoint as closed segments, i.e., no pair shares an ...

**2**

votes

**1**answer

166 views

### Lyapunov exponent for circle diffeomorphisms

Let $f:S^1\to S^1$ be an orientation-preserving circle diffeomorphism with irrational rotation number (see here). Then the system $(S^1,f)$ admits a unique invariant measure, say $\mu_f$.
Let ...

**19**

votes

**4**answers

2k views

### A question on Collatz's conjecture

Let $C$ : ${\mathbb N}\longrightarrow {\mathbb N}$ be Collatz's map defined by $C(n) = 3n+1$ if $n$ is odd, and $C(n)=n/2$ if $n$ is even. Then according to Collatz's conjecture, we should have $C^k ...

**1**

vote

**0**answers

76 views

### Uniqueness of analytic center manifold

In a book, i have read a remark which says that the center manifold of an equilibrium point of a differential equation is not unique in general but is unique in the class of analytic manifold. The ...

**1**

vote

**1**answer

147 views

### Palis' conjecture and Newhouse's results

Newhouse proved that in the space of C^r smooth diffeomorphisms r > 2, a topologically general dynamical system can have an infinite number of attractors (he goes even further, actually in showing the ...

**5**

votes

**1**answer

139 views

### Translation surfaces & integer multiples of $\pi$

Richard Schwartz, in Mostly Surfaces (Vol. 60. American Mathematical Soc., 2011),
defines (on p.14) a translation surface as "a Euclidean cone surface, all of whose 'angle errors' are integer ...

**10**

votes

**3**answers

358 views

### Which polygons have *simple* periodic billiard paths?

I know (or, rather, believe) that it remains unknown whether every polygon
has a periodic billiard path.
But Masur proved in the 1980's that every rational polygon
(vertex angles rational multiples ...

**3**

votes

**0**answers

138 views

### On the decay of correlations of an ergodic sequence over the set $X_{0}=0$

The following question arose while I was trying to explore possible further extensions of a CLT by Liverani which I mentioned here already (see this link, I can tell you more details upon request). It ...

**3**

votes

**1**answer

189 views

### invariant measure of uniquely ergodic horocycle flow

Let $S$ be a compact connected orientable surface of variable negative curvature, and let $M=T^1S$ be the unit tangent bundle of $S$. Then, we know from the paper of Brian Marcus (*) that the negative ...

**2**

votes

**0**answers

115 views

### Linear forms with best approximation vectors lying in a subspace

Setup: For $u \in \mathbb{R}^n$, let $\rho(u)$ be the Euclidean length, $\sqrt{u_1^2 + \ldots + u_n^2}$. For $x \in \mathbb{R}$ let $\|x\| = \min_{k \in \mathbb{Z}} |x - k|$, and for $x \in ...

**1**

vote

**0**answers

55 views

### A cohomology associated with a codimension one foliation(2)

What is an example of a codimension one foliation of a manifold for which this cohomology is finite dimension for all dimension $*$?
Moreover what is the description of this cohomology for ...

**1**

vote

**0**answers

195 views

### A Lie algebra associated with a one dimensional foliation

A non vanishing vector field $X$ on a manifold is called "well behaved" if for every non vanishing smooth function $f$ we have $$C(X)\simeq C(fX)$$ This means that the centralizer Lie algebras ...

**1**

vote

**1**answer

95 views

### A cohomology associated with a codimension one foliation

Let $\alpha$ be a non vanishing one form on a manifold which which defines a codimension one foliation. With this $\alpha$ we define the following complex:
$$\phi:\Omega^{i}(M)\to ...

**1**

vote

**1**answer

56 views

### positively invariant set respec to fractional system

In my research I need to show that the set
$$M := \{X \in \mathbb{R}^4,X≥0\}$$
where
$$X(t)=(x_1(t),x_2(t),x_3(t),x_4(t))^T$$
is positively invariant with respect the following system of ...

**2**

votes

**0**answers

73 views

### Analogous to a PDE but where independent variable is a function

Consider, as an example for my question, a density function $u(\boldsymbol{x},t)$ on a vector field $\boldsymbol{x}$ at some time $t$. The flow velocity vector of the density is given by ...

**14**

votes

**2**answers

275 views

### Choosing a metric in which homeomorphism is Holder continuous

Let $X$ be a compact metrizable space, and let $f:X \to X$ be a homeomorphism. Is it always possible to choose a compatible metric on $X$ in which $f$ is Holder continuous? I've tried some simple ...

**11**

votes

**2**answers

497 views

### On the boundary of the twindragon

Let $\mathcal T$ be the famous twindragon, i.e.,
$$
\mathcal T=\left\{\sum_{n=0}^\infty a_n\left(\frac{1+i}2\right)^n : a_n\in\{0,1\}\right\}.
$$
Then, as is well known, $\mathcal T$ has a ...

**0**

votes

**1**answer

98 views

### Help with notation for the state of a dynamical system defined by a PDE

Before my question let me briefly describe a simplified version of the dynamical system I'm working with. Suppose that I have a density function $m(\boldsymbol{x},t)$, that describes the abundance of ...

**1**

vote

**0**answers

54 views

### Recent Survey on Dynamics of Linear Operator

I'm studying Linear Dynamics using the textbook Linear Chaos by grosse erdmann. I'm looking for a recent encyclopaedic article/survey which gives me a big picture of the area.
It seems erdmann and ...

**3**

votes

**0**answers

94 views

### Stationarity of Brownian motion with drift

Suppose the following SDE for $X_t$ is well-posed:
$$dX_t = \sqrt{2}\, dB_t - \nabla\Phi(X_t)\,dt.$$
For what $\Phi\in C^1(R^d)$ will $X$ have stationary distribution $u_{\infty}$? For what $\Phi$ ...

**3**

votes

**1**answer

84 views

### conjugacy between geodesic flows on 2-tori

Let $(T_1,g_1)$ and $(T_2,g_2)$ be two flat tori of dimension 2 such that their geodesic flows are $C^0$-conjugated, is there an isometry between $(T_1,g_1)$ and $(T_2,g_2)$ ?
I emphasize the fact ...

**0**

votes

**0**answers

172 views

### What is the state of the art of visualizing bifurcations for “difficult” dynamical systems?

This question is related to my other recent question on MO (although I am not confident that the dynamical system described in that other question is actually "difficult," in the sense that I will ...

**4**

votes

**2**answers

231 views

### The closure of all periodic homeomorphisms of circle

Let $X$ be the space of all continuous maps from $S^{1}$ to $S^{1}$. $X$ is equipped with $C^{0}$ topology. Let $Y\subset X$ be the union of all finite order homeomorphisms i.e: all ...

**7**

votes

**1**answer

231 views

### Analytic diffeomorphisms of the circle from complex domains

Let $\gamma \subset \mathbb C$ be a simple closed analytic curve and let $\Delta$ be the closure of the disk it bounds. The Riemann mapping theorem gives two biholomorphisms:
$$\phi : (D^2,S^1) \to ...

**4**

votes

**0**answers

284 views

### A vector space associated with a vector field on a symplectic manifold

Let $(M,\omega)$ be a $2n$ dimensional symplectic manifold and $X$ is a smooth vector field on $M$. Consider the following subvector space of $\chi^{\infty}(M)$: $$S(X)=\{Y\in ...