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Stefano Galatolo

Publications and source records attributed to Stefano Galatolo.

At least 55 records · Page 3Linked to original sources
arXiv2011

Skew products, quantitative recurrence, shrinking targets and decay of correlations

We consider toral extensions of hyperbolic dynamical systems. We prove that its quantitative recurrence (also with respect to given observables) and hitting time scale behavior depend on the arithmetical properties of the extension. By this we show that those systems have a polynomial decay of correlations with respect to $C^{r}$ observables, and give estimations for its exponent, which depend on $r$ and on the arithmetical properties of the system. We also show examples of systems of this kind having not the shrinking target property, and having a trivial limit distribution of return time statistics.

math.DS↗
arXiv2011

Dynamical systems, simulation, abstract computation

We survey an area of recent development, relating dynamics to theoretical computer science. We discuss the theoretical limits of simulation and computation of interesting quantities in dynamical systems. We will focus on central objects of the theory of dynamics, as invariant measures and invariant sets, showing that even if they can be computed with arbitrary precision in many interesting cases, there exists some cases in which they can not. We also explain how it is possible to compute the speed of convergence of ergodic averages (when the system is known exactly) and how this entails the computation of arbitrarily good approximations of points of the space having typical statistical behaviour (a sort of constructive version of the pointwise ergodic theorem).

math.DS↗
arXiv2010

Computing the speed of convergence of ergodic averages and pseudorandom points in computable dynamical systems

A pseudorandom point in an ergodic dynamical system over a computable metric space is a point which is computable but its dynamics has the same statistical behavior as a typical point of the system. It was proved in [Avigad et al. 2010, Local stability of ergodic averages] that in a system whose dynamics is computable the ergodic averages of computable observables converge effectively. We give an alternative, simpler proof of this result. This implies that if also the invariant measure is computable then the pseudorandom points are a set which is dense (hence nonempty) on the support of the invariant measure.

math.NA↗
arXiv2009

Non-sequential recursive pair substitutions and numerical entropy estimates in symbolic dynamical systems

We numerically test the method of non-sequential recursive pair substitutions to estimate the entropy of an ergodic source. We compare its performance with other classical methods to estimate the entropy (empirical frequencies, return times, Lyapunov exponent). We considered as a benchmark for the methods several systems with different statistical properties: renewal processes, dynamical systems provided and not provided with a Markov partition, slow or fast decay of correlations. Most experiments are supported by rigorous mathematical results, which are explained in the paper.

cond-mat.stat-mech↗
arXiv2009

Long hitting time, slow decay of correlations and arithmetical properties

Let $τ_r(x,x_0)$ be the time needed for a point $x$ to enter for the first time in a ball $B_r(x_0)$ centered in $x_0$, with small radius $r$. We construct a class of translations on the two torus having particular arithmetic properties (Liouville components with intertwined denominators of convergents) not satisfying a logarithm law, i.e. such that for generic $x,x_0$ \liminf_{r\to 0} \frac{\log τ_r(x,x_0)}{-\log r} = \infty. By considering a suitable reparametrization of the flow generated by a suspension of this translation, using a previous construction by Fayad, we show the existence of a mixing system on three torus having the same properties. The speed of mixing of this example must be subpolynomial, because we also show that: in a system having polynomial decay of correlations the above ratio of logarithms (which is also called the lower hitting time indicator) is bounded (it is a function of the local dimension and the speed of correlation decay). More generally, this shows that reparametrizations of torus translations having a Liouville component cannot be polynomially mixing.

math.DS↗
arXiv2009

Hitting time in regular sets and logarithm law for rapidly mixing dynamical systems

We prove that if a system has superpolynomial (faster than any power law) decay of correlations (with respect to Lipschitz observables) then the time $τ(x,S_{r})$ needed for a typical point $x$ to enter for the first time a set $S_{r}=\{x:f(x)\leq r\}$ which is a sublevel of a Lipschitz funcion $f$ scales as $\frac{1}{μ(S_{r})}$ i.e. \begin{equation*} \underset{r\to 0}{\lim }\frac{\log τ(x,S_{r})}{-\log r}=\underset{r\to 0}{\lim}\frac{\log μ(S_{r})}{\log (r)}. \end{equation*} This generalizes a previous result obtained for balls. We will also consider relations with the return time distributions, an application to observed systems and to the geodesic flow of negatively curved manifolds.

math.DS↗
arXiv2009

Dynamics and abstract computability: computing invariant measures

We consider the question of computing invariant measures from an abstract point of view. We work in a general framework (computable metric spaces, computable measures and functions) where this problem can be posed precisely. We consider invariant measures as fixed points of the transfer operator and give general conditions under which the transfer operator is (sufficiently) computable. In this case, a general result ensures the computability of isolated fixed points and hence invariant measures (in given classes of "regular" measures). This implies the computability of many SRB measures. On the other hand, not all computable dynamical systems have a computable invariant measure. We exhibit two interesting examples of computable dynamics, one having an SRB measure which is not computable and another having no computable invariant measure at all, showing some subtlety in this kind of problems.

math.DS↗
arXiv2008

An elementary approach to Stochastic Differential Equations using the infinitesimals

The aim of this paper is to evidence two points relative to Nonstandard analysis (NSA): 1. In most applications of NSA to analysis, only elementary facts and techniques of nonstandard calculus seems to be necessary. 2. The advantages of a theory which includes infinitesimals rely more on the possibility of making new models rather than in the proving techniques. These two points will be illustrated in the theory of Brownian motion which can be considered as a classical model to test the power of the infinitesimal approach. Starting from a naive idea of Brownian motion, we deduce the Fokker-Plank equation in a simple and rigorous way. It is possible to keep every things to a simple level since all the theory of stochastic differential equations is treated as a hyperfinite theory and it is not translated in a "standard model". The only standard object is the final one: the Fokker-Plank equation.

math-ph↗
arXiv2008

A constructive Borel-Cantelli Lemma. Constructing orbits with required statistical properties

In the general context of computable metric spaces and computable measures we prove a kind of constructive Borel-Cantelli lemma: given a sequence (constructive in some way) of sets $A_{i}$ with effectively summable measures, there are computable points which are not contained in infinitely many $A_{i}$. As a consequence of this we obtain the existence of computable points which follow the \emph{typical statistical behavior} of a dynamical system (they satisfy the Birkhoff theorem) for a large class of systems, having computable invariant measure and a certain ``logarithmic'' speed of convergence of Birkhoff averages over Lipshitz observables. This is applied to uniformly hyperbolic systems, piecewise expanding maps, systems on the interval with an indifferent fixed point and it directly implies the existence of computable numbers which are normal with respect to any base.

math.CA↗
arXiv2008

Effective symbolic dynamics, random points, statistical behavior, complexity and entropy

We consider the dynamical behavior of Martin-Löf random points in dynamical systems over metric spaces with a computable dynamics and a computable invariant measure. We use computable partitions to define a sort of effective symbolic model for the dynamics. Through this construction we prove that such points have typical statistical behavior (the behavior which is typical in the Birkhoff ergodic theorem) and are recurrent. We introduce and compare some notions of complexity for orbits in dynamical systems and prove: (i) that the complexity of the orbits of random points equals the Kolmogorov-Sinaï entropy of the system, (ii) that the supremum of the complexity of orbits equals the topological entropy.

math.DS↗
arXiv2008

The dynamical Borel-Cantelli lemma and the waiting time problems

We investigate the connection between the dynamical Borel-Cantelli and waiting time results. We prove that if a system has the dynamical Borel-Cantelli property, then the time needed to enter for the first time in a sequence of small balls scales as the inverse of the measure of the balls. Conversely if we know the waiting time behavior of a system we can prove that certain sequences of decreasing balls satisfies the Borel-Cantelli property. This allows to obtain Borel-Cantelli like results in systems like axiom A and generic interval exchanges.

math.DS↗
arXiv2006

The recurrence time for ergodic systems of infinite measures

We investigate quantitative recurrence in systems having an infinite measure. We extend the Ornstein-Weiss theorem for a general class of infinite systems estimating return time in decreasing sequences of cylinders. Then we restrict to a class of one dimensional maps with indifferent fixed points and calculate quantitative recurrence in sequences of balls, obtaining that this is related to the behavior of the map near the fixed points.

math.DS↗
arXiv2005

Hitting time and dimension in Axiom A systems and generic interval excanges

In this note we prove that for equilibrium states of axiom A systems the time $τ_{B}(x)$ needed for a typical point $x$ to enter for the first time in a typical ball $B$ with radius $r$ scales as $τ_{B}(x)\sim r^{d}$ where $d$ is the local dimension of the invariant measure at the center of the ball. A similar relation is proved for a full measure set of interval excanges. Some applications to Birkoff averages of unbounded (and not $L^{1}$) functions are shown.

math.DS↗
arXiv2004

Algorithmic information for intermittent systems with an indifferent fixed point

Measuring the average information that is necessary to describe the behaviour of a dynamical system leads to a generalization of the Kolmogorov-Sinai entropy. This is particularly interesting when the system has null entropy and the information increases less than linearly with respect to time. We consider two classes of maps of the interval with an indifferent fixed point at the origin and an infinite natural invariant measure. We calculate that the average information that is necessary to describe the behaviour of its orbits increases with time $n$ approximately as $n^α$, where $α<1$ depends only on the asymptotic behaviour of the map near the origin.

math.DS↗
arXiv2002

Global and local Complexity in weakly chaotic dynamical systems

In a topological dynamical system the complexity of an orbit is a measure of the amount of information (algorithmic information content) that is necessary to describe the orbit. This indicator is invariant up to topological conjugation. We consider this indicator of local complexity of the dynamics and provide different examples of its behavior, showing how it can be useful to characterize various kind of weakly chaotic dynamics. We also provide criteria to find systems with non trivial orbit complexity (systems where the description of the whole orbit requires an infinite amount of information). We consider also a global indicator of the complexity of the system. This global indicator generalizes the topological entropy, taking into account systems were the number of essentially different orbits increases less than exponentially. Then we prove that if the system is constructive (roughly speaking: if the map can be defined up to any given accuracy using a finite amount of information) the orbit complexity is everywhere less or equal than the generalized topological entropy. Conversely there are compact non constructive examples where the inequality is reversed, suggesting that this notion comes out naturally in this kind of complexity questions.

math.DS↗
arXiv2001

Information, initial condition sensitivity and dimension in weakly chaotic dynamical systems

We study generalized indicators of sensitivity to initial conditions and orbit complexity in topological dynamical systems. The orbit complexity is a measure of the asymptotic behavior of the information that is necessary to describe the orbit of a given point. The indicator generalizes in a certain sense, the Brudno's orbit complexity (which is strongly related to the entropy of the system). The initial condition sensitivity indicators consider the asymptotic behavior of the speed of divergence of nearby staring orbits. The indicators have non trivial values also in weakly chaotic dynamical systems, characterizing various cases of weakly chaotic dynamics. Then, using constructivity, local relations are proved between generalized orbit complexity, initial condition sensitivity and the dimension of the underlying space or invariant measure

math.DS↗
arXiv2001

Information, complexity and entropy: a new approach to theory and measurement methods

In this paper, we present some results on information, complexity and entropy as defined below and we discuss their relations with the Kolmogorov-Sinai entropy which is the most important invariant of a dynamical system. These results have the following features and motivations: -we give a new computable definition of information and complexity which allows to give a computable characterization of the K-S entropy; -these definitions make sense even for a single orbit and can be measured by suitable data compression algorithms; hence they can be used in simulations and in the analysis of experimental data; -the asymptotic behavior of these quantities allows to compute not only the Kolmogorov-Sinai entropy but also other quantities which give a measure of the chaotic behavior of a dynamical system even in the case of null entropy.

math.DS↗