The essence of Dynamical Systems is to mathematically understand general laws governing processes undergoing transformations on time. Amongst its main concerns, one wants to decide whether an asymptotic behaviour of the dynamics is robust under small random fluctuations, or random perturbations. It represents a natural point of view since observations of phenomena in nature are always subjected to small fluctuations. Although the interest in random perturbation of dynamical systems goes back to Kolmogorov, its systematic study is relatively recent. The methods used to tackle such problems demand and stimulate a rich exchange between different mathematical areas such as Probability, Analysis, Geometry, etc.

The aim of the workshop is to bring together experts working on different aspects related to random perturbations and statistical properties of dynamical systems, as well as young researchers and PhD students interested in the area.

All the speakers are supposed to motivate their talks and embed them into a broad context, in order to make them accessible also to non-specialists. In addition, there will be a few survey lectures on recent advances and techniques used to describe from a statistical point of view randomly perturbed dynamics, and a few contributed talks by young researchers.

The topics of the conference include:

random perturbations and physical measures of dynamical systems;

large deviations and concentration inequalities;

stochastic aspects of bifurcations;

statistical and stochastic stability.

Speakers

Jean-René Chazottes

École Polytechnique, France

Jorge M. M. de Freitas

Faculdade de Ciências da Universidade do Porto, Portugal

Franco Flandoli

Università di Pisa, Italy

Barbara Gentz

Universität Bielefeld, Germany

Ale Jan Homburg

University of Amsterdam, Netherlands

Yuri I. Kifer

The Hebrew University of Jerusalem, Israel

Paulo R. C. Ruffino

Universidade Estadual de Campinas, Brazil and Humboldt-Universität zu Berlin, Germany

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.

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 $\epsilon$ 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 $\epsilon$ 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.

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 $\mathbb{R}^n$. 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).

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).

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).

I will start with a description of results on various limit theorems for ''nonconventional'' sums of the form $\sum_{n=1}^{N}B(\xi(q_{1}(n)), \xi(q_{2}(n)), \ldots, \xi(q_{\ell} (n)))$, where $\xi(n)$ for some $n \geq 0$ is some stochastic process or a dynamical system while $q_{i}(n)$ are integer valued and are linear for $i \leq k$ and grow faster than linearly for $i > k , i \leq \ell$. 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 $\Omega$ 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).

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.

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.

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.

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.

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 L1-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.

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.

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)

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.

Participants

José Ferreira Alves

University of Porto, Portugal

Hale Aytaç

University of Porto, Portugal

Imran Habib Biswas

Tata Institute of Fundamental Research, India

Doris Bohnet

Université de Bourgogne, France

Jean-René Chazottes

École Polytechnique, France

Jorge M. M. de Freitas

Faculdade de Ciências da Universidade do Porto, Portugal

Manh Hong Duong

Eindhoven University of Technology, Netherlands

Albertas Dvirnas

Kaunas University of Technology, Lithuania

Peyman Eslami

University of Rome "Tor Vergata", Italy

Franco Flandoli

Università di Pisa, Italy

Manjunath Gandhi

Jacobs University Bremen, Germany

Barbara Gentz

Universität Bielefeld, Germany

Ale Jan Homburg

University of Amsterdam, Netherlands

Jürgen Jost

Max Planck Institute for Mathematics in the Sciences, Germany

Thomas Kaijser

Linkoping University, Sweden

Martin Kell

Max Planck Institute for Mathematics in the Sciences, Germany

Yuri I. Kifer

The Hebrew University of Jerusalem, Israel

Rüdiger Kürsten

Max Planck Institute for Mathematics in the Sciences, Germany

Tobias Lehmann

University of Bath, Germany

Ralph Lettau

University of Augsburg, Germany

Stefano Luzzatto

Abdus Salam International Centre for Theoretical Physics - ICTP, Italy

Ziyad Muslimani

Universität Düsseldorf, Germany

Julian Newman

Imperial College London, United Kingdom

Thu Nguyen

University of Bonn, Germany

Vasilis Oikonomou

Max Planck Institute for Mathematics in the Sciences, Greece

Eckehard Olbrich

Max Planck Institute for Mathematics in the Sciences, Germany

Atsuya Otani

Universität Erlangen, Germany

Jing Qin

Interdisziplinary Center for Bioinformatics, Germany

Johannes Rauh

MPI MIS, Germany

Christian S. Rodrigues

Max Planck Institute for Mathematics in the Sciences, Germany

Lavinia Roncoroni

Max Planck Institute, Germany

Paulo R. C. Ruffino

Universidade Estadual de Campinas, Brazil

Christian Seifert

TU Hamburg-Harburg, Germany

Upanshu Sharma

TU/e, Netherlands

Sacha Sokoloski

Max Planck Institute for Mathematics in the Sciences, Germany

Emanuele Spadaro

Max Planck Institute for Mathematics in the Science, Germany

Doan Thai Son

Imperial College London, United Kingdom

Tat Dat Tran

Max Planck Institute for Mathematics in the Sciences, Germany

Sandro Vaienti

Centre de Physique Théorique - Luminy, France

Alison Marcelo Van Der Laan Melo

UNICAMP, Brazil

Helder Vilarinho

Universidade da Beira Interior, Portugal

Shouhong Wang

Indiana University, USA

Ran Wang

University of Amsterdam, Netherlands

Wei Yang

RICAM, Austrain Academy of Sciences, Austria

Odette Sylvia Yaptieu Djeungue

Max-Planck-Institute for Mathematics in the Sciences, Germany

Scientific Organizers

José Ferreira Alves

University of Porto, 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, Germany
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