# Random Perturbations and Statistical Properties of Dynamical Systems

## Abstracts

**José Ferreira Alves***University of Porto, Portugal***Statistical stability in chaotic dynamics **

We consider families of chaotic dynamical systems possessing physical measures which describe the statistical behavior of typical orbits. We study the continuous dependence of the physical measure with respect to transformation or flow that rules the dynamical system. Some general results will be given. Quadratic maps of the interval and Lorenz flows will be considered in particular.

**Hale Aytaç***University of Porto, Portugal***Extremal behaviour of randomly perturbed dynamical systems**

In this talk, we will explain how we get laws of rare events for randomly perturbed dynamical systems using the link between Extreme Value Laws (EVL) and Hitting/Return Time Statistics (HTS/RTS). Mainly, we will consider random perturbations of uniformly expanding systems, such as piecewise expanding maps of the circle, and show that for additive absolutely continuous (w.r.t. Lebesgue) noise, the limiting distribution is standard exponential for any point. Our main ingredient will be decay of correlations against all L^{1} observables in a suitable Banach space and due to the above link we get our results by means of the first return time from a set to itself.

**Doris Bohnet***Université de Bourgogne, France***Margulis measure in the context of partially hyperbolic dynamics **

Let f be a diffeomorphism on a compact manifold M and assume that the tangent bundle splits into three subbundles stable, unstable and central - invariant under df such that df contracts a vector of the stable bundle, expands a vector of the unstable bundle and finally contracts/ expands to a weaker degree a vector of the central bundle. Such a diffeomorphism is called partially hyperbolic. We consider partially hyperbolic diffeomorphisms where the central bundle integrates to a compact central foliation, that is every center leaf is a compact manifold. We describe the structure and properties of these diffeomorphisms and establish the existence of a Margulis measure in this context (under some additional assumptions).

**Jean-René Chazottes***École Polytechnique, France***On concentration inequalities for dynamical systems **

After a broad introduction, I will talk about stochastic processes generated by dynamical systems. Depending on the speed of decay of correlations, one gets exponential or polynomial concentration inequalities for a large class of observables.

**Jorge M. M. de Freitas***Faculdade de Ciências da Universidade do Porto, Portugal***Extremal behaviour of deterministic and random dynamical systems**

It is well known that the Extremal Index (EI) measures the intensity of clustering of extreme events in stationary processes. We sill see that for some certain uniformly expanding systems there exists a dichotomy based on whether the rare events correspond to the entrance in small balls around a periodic point or a non-periodic point. In fact, either there exists EI in *(0,1)* around (repelling) periodic points or the EI is equal to *1* at every non-periodic point. The main assumption is that the systems have sufficient decay of correlations of observables in some Banach space against all *L ^{1}*-observables.

Then we consider random perturbations of uniformly expanding systems, such as piecewise expanding maps of the circle. We will see that, in this context, for additive absolutely continuous noise (w.r.t. Lebesgue), the dichotomy vanishes and the EI is always 1.

**Franco Flandoli***Università di Pisa, Italy***Is the random attractor a singleton?**

In analogy with deterministic theory, one initially expects that random attractors may be complex objects, but in a number of examples where the corresponding deterministic dynamics has a non-trivial attractor it has been proved that the random attractor is indeed a random singleton. This phenomenon is related to stochastic bifurcation and its different interpretations. One-point random attractors seem to be more common than expected; but there are also examples where it is not one point. We shall review a few old and new facts about this topic, and related issues as the properties of the two-point motion.

**Barbara Gentz***Universität Bielefeld, Germany***Small eigenvalues and mean transition times for irreversible diffusions **

The spectral theory of reversible diffusions in the small-noise limit is well understood. The small eigenvalues of the generator have been analyzed by a number of different methods, including large deviations, semiclassical analysis and potential theory. However, the study of the irreversible case, which involves a non-selfadjoint generator, is substantially more difficult. We will discuss an approach based on Laplace transforms of hitting times for Markov chains with continuous state space. These Markov chains arise from random Poincaré maps.

The proposed approach provides information on the exponentially small eigenvalues of the generator, and on mean transition times between attractors. As an illustration, we will present a detailed analysis of the asymptotic behaviour of the first-passage time of a planar diffusion through an unstable periodic orbit in the small-noise limit.

Joint work with Nils Berglund (Orléans).

**Ale Jan Homburg***University of Amsterdam, Netherlands***Iterated function systems, skew product dynamics, partially hyperbolic dynamics**

I'll consider iterated function systems generated by finitely many diffeomorphisms on compact manifolds. I'll discuss aspects of their dynamics, in particular minimality and synchronization.

These iterated function systems play a pivotal role in the study of dynamical systems: they correspond to dynamical systems of skew product type and provide examples of "partially hyperbolic dynamical systems". I'll discuss how iterated function systems are giving new results and insights in the study of partially hyperbolic dynamics.

**Martin Kell***Max Planck Institute for Mathematics in the Sciences, Germany***Representation of Markov chains via random maps **

In the theory of random perturbations two of the main modelling objects are random maps and Markov chains. It is easy to show that every random map driven by a Markov process can represented by a Markov chain. The opposite is less clear. Previous approaches assumed smooth and sometimes even uniform noise, and implicitly required parallelizability of the tangent space. In this talk we attack this problem translating it into the language of optimal transport theory. Via some well-known lifts to trivial vector bundles we show that the problem can be reduced to study parametrized families of measure in ℝ^{n}. An existence proof for measurable random maps directly follows from the existence of transport maps and their stability properties. Using some more recent regularity results, it can be shown that a Markov chain satisfying some mild assumptions can be represented by a continuous random map (Joint work with Jost, and Rodrigues).

**Yuri I. Kifer***The Hebrew University of Jerusalem, Israel***Nonconventional limit theorems**

I will start with a description of results on various limit theorems for ”nonconventional” sums of the form ∑_{n=1}^{N}B(ξ(q_{1}(n)),ξ(q_{2}(n)),…,ξ(q_{ℓ}(n))), where ξ(n) for some n ≥ 0 is some stochastic process or a dynamical system while q_{i}(n) are integer valued and are linear for i ≤ k and grow faster than linearly for i > k,i ≤ ℓ. The motivation for this study comes, in particular, from many papers about nonconventional ergodic theorems appeared in the last 30 years. Such limit theorems describe multiple recurrence properties of the corresponding processes and dynamical systems and they include central limit theorem, large deviations, averaging which will be discussed in the first part of my talk and Poisson type limit theorems which will form the second part of my talk.

During last 20 years a substantial attention was attracted to the study of numbers of arrivals at small (shrinking) sets by trajectories of dynamical systems during the time intervals inversely proportional to the measure of a set. It seems that the question was originated by Sinai in the framework of the study of distributions of spacings between energy levels of quantum systems. Most of the papers dealt with the symbolic setup of a sequence space Ω with a shift invariant sufficiently fast mixing probability P where Pitskel, Hirata and Denker showed that when P is a Gibbs shift invariant measure then the numbers of arrivals to shrinking cylindrical neighborhoods of almost all points are asymptotically Poisson distributed. More recently estimates for Poisson approximations were obtained by Abadi and others while Haidn and Vaienti obtained compound Poisson approximations for distributions of numbers of arrivals to shrinking cylindrical neighborhoods of some periodic points. A year and a half ago Poisson limiting behavior in the symbolic situation was extended to numbers of multiple recurrencies (nonconventional setup). In this talk I provide an essentially complete description of possible limiting behaviors of distributions of numbers of multiple recurrencies to shrinking cylinders which is mostly new even for the widely studied single (conventional) recurrencies case (joint work with my student Ariel Rapaport).

**Atsuya Otani***Universität Erlangen, Germany***Bifurcation and Hausdorff dimension in families of chaotically driven maps with multiplicative forcing**

We consider the bifurcation of invariant graphs in a parametrized skew-product system with hyperbolic base-map.

In contrast to quasi-periodically driving systems, the hyperbolic driving causes a smooth bifurcation process from a point of view of the dimension theory.

We mainly discuss this phenomenon using the thermodynamic formalisms. (Joint work with Gerhard Keller)

**Paulo R. C. Ruffino***Universidade Estadual de Campinas, Brazil, and Humboldt-Universität zu Berlin, Germany***Averaging along diffusions in foliated manifolds**

Consider a diffusion in a foliated manifold whose trajectories lay on compact leaves. We investigate the effective behaviour of a small transversal perturbation of order ϵ which destroys the foliated trajectories. An average principle is shown to hold such that the transversal component to the leaves converges to the solution of a deterministic ODE, according to the average of the perturbing vector field with respect to invariant measures on the leaves of the original foliated system, as ϵ goes to zero. An estimate of the rate of convergence is given. These results generalize the geometrical scope of previous approaches, including completely integrable stochastic Hamiltonian system. The diffusion can be generated either by Stratonovich SDE, Lévy noise (Marcus equation) and others. These results, with different types of diffusions, have been obtained in colaborations with Michael Högele, I. Gonzales-Gargate and P. H. da Costa.

**Sandro Vaienti***Centre de Physique Théorique - Luminy, France***Extremes and Borel-Cantelli theorems for randomly perturbed systems**

We present a few new results on extreme value theory applied to systems perturbed with observational noise. Moreover we review recent results on the Borel-Cantelly property for dynamical systems pertubed "via" random transformations.

**Helder Vilarinho***Universidade da Beira Interior, Portugal***Decay of correlations for random non-uniformly expanding maps**

We discuss the decay of correlations along orbits generated by random perturbations of general non-uniformly expanding maps. We present results for different rates - polynomial and (stretched) exponential - according to the hypotheses on the decay of the measure of the 'tail sets'. As applications, we get new estimates for this random decay of correlations for some families of maps (e.g., the Viana maps) and we also re-obtain some known results, sometimes with improvements on the rates, as is the case of the quadratic family.

This is a joint work (in progress) with Xin Li.

## Date and Location

**July 09 - 11, 2013**

Max Planck Institute for Mathematics in the Sciences

Inselstraße 22

04103 Leipzig

Germany

see travel instructions

## Scientific Organizers

**José Ferreira Alves**

University of Porto

Department of Mathematics

Portugal

**Jürgen Jost**

Max Planck Institute for Mathematics in the Sciences

Germany

**Stefano Luzzatto**

Abdus Salam International Centre for Theoretical Physics - ICTP

Italy

**Christian S. Rodrigues**

Max Planck Institute for Mathematics in the Sciences

Germany

## Administrative Contact

**Antje Vandenberg**

Max Planck Institute for Mathematics in the Sciences

Contact by Email