Related papers: A note on Eckmann-Ruelle's conjecture

Let $\lambda$ be a probability measure on $\mathbb T^{n-1}$ where $n=2$ or 3. Suppose $\lambda$ is invariant, ergodic and has positive entropy with respect to the linear transformation defined by a hyperbolic matrix. We get a measure $\mu $…

For a continuous Hamiltonian $H : (x, p, u) \in T^*\mathbb{R}^n \times \mathbb{R}\rightarrow \mathbb{R}$, we consider the asymptotic behavior of associated Hamilton--Jacobi equations with state-constraint $H(x, Du, \lambda u) \leq…

Let $1<\beta \leq 2$. It is well-known that the set of points in $% [0,1/(\beta -1)]$ having unique $\beta $-expansion, in other words, those points whose orbits under greedy $\beta $-transformation escape a hole depending on $\beta $, is…

Let $\Omega\subset\mathbb R^2$ be a bounded domain of class $C^{2+\alpha}$, $0<\alpha<1$. We show that if $u$ is the solution of $\Delta u = 4\exp(2u)$ which tends to $+\infty$ as $(x,y)\to\partial\Omega$, then the hyperbolic radius…

Let $E$ be a finite set, $\{F^i\}_{i \in E}$ a family of vector fields on $\mathbb{R}^d$ leaving positively invariant a compact set $M$ and having a common zero $p \in M.$ We consider a piecewise deterministic Markov process $(X,I)$ on $M…

We study $C^1$-generic vector fields on closed manifolds without points accumulated by periodic orbits of different indices and prove that they exhibit finitely many sinks and sectional-hyperbolic transitive Lyapunov stable sets with…

We consider products of a i.i.d. sequence in a set $\{f_1,\ldots,f_m\}$ of preserving orientation diffeomorphisms of the circle. we can naturally associate a Lyapunov exponent $\lambda$. Under few assumptions, it is known that $\lambda\leq…

We show that dynamical systems with $\phi$-mixing measures have local escape rates which are exponential with rate $1$ at non-periodic points and equal to the extremal index at periodic points. We apply this result to equilibrium states on…

For the continuous-time $\lambda$-recurrent jump process, the $\lambda$-recurrence assures the existence of quasi-stationary distribution when it has finite exit states (the states that have positive killing rates). And we give an explicit…

We consider the linear stochastic recursion $x_{i+1} = a_{i}x_{i}+b_{i}$ where the multipliers $a_i$ are random and have Markovian dependence given by the exponential of a standard Brownian motion and $b_{i}$ are i.i.d. positive random…

We study the convergence rate of Sinkhorn's algorithm for solving entropy-regularized optimal transport problems when at least one of the probability measures, $\mu$, admits a density over $\mathbb{R}^d$. For a semi-concave cost function…

The fundamental inequality of Guivarc'h relates the entropy and the drift of random walks on groups. It is strict if and only if the random walk does not behave like the uniform measure on balls. We prove that, in any nonelementary…

Let S_n=X_1+...+X_n be a sum of independent symmetric random variables such that |X_{i}|\leq 1. Denote by W_n=\epsilon_{1}+...+\epsilon_{n} a sum of independent random variables such that \prob{\eps_i = \pm 1} = 1/2. We prove that…

In this paper we examine the task set forth by Shelah and Villaveces in \cite{ShVi} of proving the uniqueness of limit models of cardinality $\mu$ in $\lambda$-categorical abstract elementary classes with no maximal models, where $\lambda$…

Solutions to the random Fibonacci recurrence x_{n+1}=x_{n} + or - Bx_{n-1} decrease (increase) exponentially, x_{n} = exp(lambda n), for sufficiently small (large) B. In the limits B --> 0 and B --> infinity, we expand the Lyapunov exponent…

It has long been conjectured that the classical Lorenz attractor supports a unique measure of maximal entropy. In this article, we give a positive answer to this conjecture and its higher-dimensional counterpart by considering the…

We prove a $C^1$ version of a conjecture by Pugh and Shub: among partially hyperbolic volume-preserving $C^r$ diffeomorphisms, $r>1$, the stably ergodic ones are $C^1$-dense. To establish these results, we develop new perturbation tools for…

We study the asymptotic response of polar ordered active fluids ("flocks") to small external aligning fields $h$. The longitudinal susceptibility $\chi_{_\parallel}$ diverges, in the thermodynamic limit, like $h^{-\nu}$ as $h \rightarrow…

For a class of volume preserving partially hyperbolic diffeomorphisms (or non-uniformly Anosov) $f\colon {\T}^d\rightarrow{\T}^d$ homotopic to linear Anosov automorphism, we show that the sum of the positive (negative) Lyapunov exponents of…

We prove that the Bochi-Ma\~{n}\'{e} theorem is false, in general, for linear cocycles over non-invertible maps: there are C0-open subsets of linear cocycles that are not uniformly hyperbolic and yet have Lyapunov exponents bounded from…