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Register a new userBirkhoff sums as distributions II: Applications to deformations of dynamical systems
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Often topological classes of one-dimensional dynamical systems are finite codimension smooth manifolds. We describe a method to prove this sort of statement that we believe can be applied in many settings. In this work we will implement it for piecewise expanding maps. The most important step will be the identification of infinitesimal deformations with primitives of Birkhoff sums (up to addition of a Lipschitz function), that allows us to use the ergodic properties of piecewise expanding maps to study the regularity of infinitesimal deformations.
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We study Birkhoff sums as distributions. We obtain regularity results on such distributions for various dynamical systems with hyperbolicity, as hyperbolic linear maps on the torus and piecewise expanding maps on the interval. We also give some applications, as the study of advection in discrete dynamical systems.
We establish two precise asymptotic results on the Birkhoff sums for dynamical systems. These results are parallel to that on the arithmetic sums of independent and identically distributed random variables previously obtained by Hsu and Robbins, ErdH{o}s, Heyde. We apply our results to the Gauss map and obtain new precise asymptotics in the theorem of Levy on the regular continued fraction expansion of irrational numbers in $(0,1)$.
We establish quantitative results for the statistical be-ha-vi-our of emph{infinite systems}. We consider two kinds of infinite system: i) a conservative dynamical system $(f,X,mu)$ preserving a $sigma$-finite measure $mu$ such that $mu(X)=infty$; ii) the case where $mu$ is a probability measure but we consider the statistical behaviour of an observable $phicolon Xto[0,infty)$ which is non-integrable: $int phi , dmu=infty$. In the first part of this work we study the behaviour of Birkhoff sums of systems of the kind ii). For certain weakly chaotic systems, we show that these sums can be strongly oscillating. However, if the system has superpolynomial decay of correlations or has a Markov structure, then we show this oscillation cannot happen. In this case we prove asymptotic relations between the behaviour of $phi $, the local dimension of $mu$, and on the growth of Birkhoff sums (as time tends to infinity). We then establish several important consequences which apply to infinite systems of the kind i). This includes showing anomalous scalings in extreme event limit laws, or entrance time statistics. We apply our findings to non-uniformly hyperbolic systems preserving an infinite measure, establishing anomalous scalings in the case of logarithm laws of entrance times, dynamical Borel--Cantelli lemmas, almost sure growth rates of extremes, and dynamical run length functions.
This paper is aimed at a detailed study of the multifractal analysis of the so-called divergence points in the system of $beta$-expansions. More precisely, let $([0,1),T_{beta})$ be the $beta$-dynamical system for a general $beta>1$ and $psi:[0,1]mapstomathbb{R}$ be a continuous function. Denote by $textsf{A}(psi,x)$ all the accumulation points of $Big{frac{1}{n}sum_{j=0}^{n-1}psi(T^jx): nge 1Big}$. The Hausdorff dimensions of the sets $$Big{x:textsf{A}(psi,x)supset[a,b]Big}, Big{x:textsf{A}(psi,x)=[a,b]Big}, Big{x:textsf{A}(psi,x)subset[a,b]Big}$$ i.e., the points for which the Birkhoff averages of $psi$ do not exist but behave in a certain prescribed way, are determined completely for any continuous function $psi$.
This paper is devoted to study multifractal analysis of quotients of Birkhoff averages for countable Markov maps. We prove a variational principle for the Hausdorff dimension of the level sets. Under certain assumptions we are able to show that the spectrum varies analytically in parts of its domain. We apply our results to show that the Birkhoff spectrum for the Manneville-Pomeau map can be discontinuous, showing the remarkable differences with the uniformly hyperbolic setting. We also obtain results describing the Birkhoff spectrum of suspension flows. Examples involving continued fractions are also given.
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