Tagged Questions

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

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Unique Nash equilibrium games

Multicast network design game is a special case of a general network design game (http://www.cs.cornell.edu/home/kleinber/focs04-game.pdf) in which there is a target vertex $t$ and $n$ rational ...
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Convergence to equilibrium for time in-homogeneous diffusions

Consider the long time behavior for a time in-homogeneous diffusion such as $$dX_t = dB_t - \nabla V(X_t)\,dt + b_t(X_t)dt,$$ where $V(x)$ is a smooth convex function and $b_t(x)$ is a time-dependent ...
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Is there a universal $\omega$-limit set?

For the purposes of this question, a dynamical system means a compact metric space $X$ together with a continuous map $f: X \to X$. For $x \in X$, the $\omega$-limit set of $x$, denoted $\omega(x)$, ...
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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 $$||\phi||_{L^\... 3answers 246 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 ... 2answers 272 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 ... 0answers 103 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 ... 2answers 165 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 ... 1answer 144 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 ... 2answers 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 ... 1answer 198 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 ... 1answer 183 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 ... 1answer 133 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 ... 0answers 117 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}} f(\... 1answer 527 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 ... 4answers 628 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, ... 0answers 337 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 ... 2answers 199 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 ... 1answer 115 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 ... 2answers 115 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 A_1X_0.$... 0answers 609 views Time for Langton's ant to cover a “square” torus Langton's ant is a cellular automaton running as follows: Squares on a plane are colored variously either black or white. We arbitrarily identify one square as the "ant". The ant can travel ... 1answer 174 views length comparison on negatively curved surfaces Suppose$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}$the set of free homotopy ... 1answer 882 views In how many steps a random walk visits all the elements of a finite group, with a probability 1/2? This question is a variation of the return to the origin problem. Let$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) = (...
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 ...
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 $M\times M$, ...