No Arabic abstract
Given Holder continuous functions $f$ and $psi$ on a sub-shift of finite type $Sigma_A^{+}$ such that $psi$ is not cohomologous to a constant, the classical large deviation principle holds (cite{OP}, cite{Kif}, cite{Y}) with a rate function $I_psigeq 0$ such that $I_psi (p) = 0$ iff $p = int psi , d mu$, where $mu = mu_f$ is the equilibrium state of $f$. In this paper we derive a uniform estimate from below for $I_psi$ for $p$ outside an interval containing $tilde{psi} = int psi , dmu$, which depends only on the sub-shift, the function $f$, the norm $|psi|_infty$, the Holder constant of $psi$ and the integral $tilde{psi}$. Similar results can be derived in the same way e.g. for Axiom A diffeomorphisms on basic sets.
For an arbitrary negative Schwarzian unimodal map with non-flat critical point, we establish the level-2 Large Deviation Principle (LDP) for empirical distributions. We also give an example of a multimodal map for which the level-2 LDP does not hold.
For hyperbolic flows $varphi_t$ we examine the Gibbs measure of points $w$ for which $$int_0^T G(varphi_t w) dt - a T in (- e^{-epsilon n}, e^{- epsilon n})$$ as $n to infty$ and $T geq n$, provided $epsilon > 0$ is sufficiently small. This is similar to local central limit theorems. The fact that the interval $(- e^{-epsilon n}, e^{- epsilon n})$ is exponentially shrinking as $n to infty$ leads to several difficulties. Under some geometric assumptions we establish a sharp large deviation result with leading term $C(a) epsilon_n e^{gamma(a) T}$ and rate function $gamma(a) leq 0.$ The proof is based on the spectral estimates for the iterations of the Ruelle operators with two complex parameters and on a new Tauberian theorem for sequence of functions $g_n(t)$ having an asymptotic as $ n to infty$ and $t geq n.$
We give an exponential upper bound on the probabilitywith which the denominator of the $n$th convergent in the regular continued fraction expansion stays away from the mean $frac{npi^2}{12log2}$. The exponential rate is best possible, given by an analytic function related to the dimension spectrum of Lyapunov exponents for the Gauss transformation.
We study equidistribution properties of translations on nilmanifolds along functions of polynomial growth from a Hardy field. More precisely, if $X=G/Gamma$ is a nilmanifold, $a_1,ldots,a_kin G$ are commuting nilrotations, and $f_1,ldots,f_k$ are functions of polynomial growth from a Hardy field then we show that $bullet$ the distribution of the sequence $a_1^{f_1(n)}cdotldotscdot a_k^{f_k(n)}Gamma$ is governed by its projection onto the maximal factor torus, which extends Leibmans Equidistribution Criterion form polynomials to a much wider range of functions; and $bullet$ the orbit closure of $a_1^{f_1(n)}cdotldotscdot a_k^{f_k(n)}Gamma$ is always a finite union of sub-nilmanifolds, which extends some of the previous work of Leibman and Frantzikinakis on this topic.
We construct a stochastic model showing the relationship between noise, gradient flows and rate-independent systems. The model consists of a one-dimensional birth-death process on a lattice, with rates derived from Kramers law as an approximation of a Brownian motion on a wiggly energy landscape. Taking various limits we show how to obtain a whole family of generalized gradient flows, ranging from quadratic to rate-independent ones, connected via $L log L$ gradient flows. This is achieved via Mosco-convergence of the renormalized large-deviations rate functional of the stochastic process.