Statistical properties of dynamics. Introduction to the functional analytic approach

These are lecture notes for a simple minicourse approaching the satistical properties of a dynamical system by the study of the associated transfer operator (considered on a suitable function space). The following questions will be addressed: * existence of a regular invariant measure; * Lasota Yorke inequalities and spectral gap; * decay of correlations and some limit theorem; * stability under perturbations of the system; * linear response; * hyperbolic systems. The point of view taken is to present the general construction and ideas needed to obtain these results in the simplest way. For this, some theorem is proved in a form which is weaker than usually known, but with an elementary and simple proof. These notes are intended for the Hokkaido-Pisa University summer course 2017.Comment: I decided to make these lecture notes public because it will be cited in some research paper. I hope these will be useful for some reader. In this new version several new topics are added, with some original approac

Quantitative statistical stability and convergence to equilibrium. An application to maps with indifferent fixed points

We show a general relation between fixed point stability of suitably perturbed transfer operators and convergence to equilibrium (a notion which is strictly related to decay of correlations). We apply this relation to deterministic perturbations of a large class of maps with indifferent fixed points. It turns out that the $L^1$ dependence of the a.c.i.m. on small suitable deterministic changes for these kind of maps is H\"older, with an exponent which is explicitly estimated.Comment: Second revision with some improvemen

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.Comment: 23 page

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.Comment: In this revision some small errors are correcte

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 $\tau (x,S_{r})$ 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 $\frac{1}{\mu (S_{r})}$ i.e. \begin{equation*} \underset{r\to 0}{\lim }\frac{\log \tau (x,S_{r})}{-\log r}=\underset{r\to 0}{\lim}\frac{\log \mu (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