Isaia Nisoli

An Elementary Approach to Rigorous Approximation of Invariant Measures

We describe a framework in which it is possible to develop and implement algorithms for the approximation of invariant measures of dynamical systems with a given bound on the error of the approximation. Our approach is based on a general statement on the approximation of fixed points for operators between normed vector spaces, allowing an explicit estimation of the error. We show the flexibility of our approach by applying it to piecewise expanding maps and to maps with indifferent fixed points. We show how the required estimations can be implemented to compute invariant densities up to a given error in the

Rigorous computation of invariant measures and fractal dimension for piecewise hyperbolic maps: 2D Lorenz like maps.

L^{1}

Shrinking targets in fast mixing flows and the geodesic flow on negatively curved manifolds

or

Rigorous approximation of stationary measures and convergence to equilibrium for iterated function systems

L^\infty

Decay of Correlations, Quantitative Recurrence and Logarithm Law for Contracting Lorenz Attractors

distance. We also show how to use this to compute an estimation with certified error for the entropy of those systems. We show how several related computational and numerical issues can be solved to obtain working implementations and experimental results on some one dimensional maps.

Probability, statistics and computation in dynamical systems

We consider a class of piecewise hyperbolic maps from the unit square to itself preserving a contracting foliation and inducing a piecewise expanding quotient map, with innite derivative (like the rst return maps of Lorenz like ows). We show how the physical measure of those systems can be rigorously approximated with an explicitly given bound on the error, with respect to the Wasserstein distance. We apply this to the rigorous computation of the dimension of the measure. We present a rigorous implementation of the algorithms using interval arithmetics, and the result of the computation on a nontrivial example of Lorenz like map and its attractor, obtaining a statement on its local dimension.

A rigorous computational approach to linear response

We show that in a rapidly mixing flow with an invariant measure, the time which is needed to hit a given section is related to a sort of conditional dimension of the measure at the section. The result is applied to the geodesic flow of compact variable negative sectional curvature manifolds, establishing a logarithm law for such kind of flow.

An elementary way to rigorously estimate convergence to equilibrium and escape rates

We study the problem of the rigorous computation of the stationary measure and of the rate of convergence to equilibrium of an iterated function system described by a stochastic mixture of two or more dynamical systems that are either all uniformly expanding on the interval, either all contracting. In the expanding case, the associated transfer operators satisfy a Lasota–Yorke inequality, we show how to compute a rigorous approximations of the stationary measure in the L 1 norm and an estimate for the rate of convergence. The rigorous computation requires a computer-aided proof of the contraction of the transfer operators for the maps, and we show that this property propagates to the transfer operators of the IFS. In the contracting case we perform a rigorous approximation of the stationary measure in the Wasserstein–Kantorovich distance and rate of convergence, using the same functional analytic approach. We show that a finite computation can produce a realistic computation of all contraction rates for the whole parameter space. We conclude with a description of the implementation and numerical experiments.

Rigorous computation of invariant measures and fractal dimension for maps with contracting fibers: 2D Lorenz-like maps

In this paper we prove that a class of skew products maps with non uniformly hyperbolic base has exponential decay of correlations. We apply this to obtain a logarithm law for the hitting time associated to a contracting Lorenz attractor at all the points having a well defined local dimension, and a quantitative recurrence estimation.

Rigorous Approximation of Diffusion Coefficients for Expanding Maps

We discuss some recent results related to the deduction of a suitable probabilistic model for the description of the statistical features of a given deterministic dynamics. More precisely, we motivate and investigate the computability of invariant measures and some related concepts. We also present some experiments investigating the limits of naive simulations in dynamics.