Monday Tuesday Wednesday Thursday Friday 9:00-9:30 9:00-11:00 9:00-11:00 9:00-11:00 9:00-11:00 Zhihong Xia Open remark Jose Alves Jose Alves Nikolai Haydn Richard Sharp 9:30 - 11:00 Huyi Hu 2:00-3:00 2:00-4:00 2:00-4:00 2:00-4:00 2:00-4:00 Jianyu Chen Jianyu Chen Nikolai Haydn Richard Sharp Nikolai Haydn 3:20-5:20 4:20-5:20 4:20-5:20 4:20-5:20 Jose Alves Huyi Hu Jianyu Chen Huyi Hu

Monday Tuesday Wednesday Thursday Friday 8：30-11：00 8:00-10:00 8:00-10:00 8:00-10:00 8:00-10:00 Hongkun Zhang Hongkun Zhang Mark Pollicott Carlangelo Liverani Sandro Vaienti 10:20-11:20 10:20-11:20 10:20-11:20 10:20-11:20 Mark Pollicott Carlangelo Liverani Sandro Vaienti Dmitry Dolgopyat 2:00-3:00 2:00-3:00 2:00-3:00 2:00-3:00 2:00-3:00 Hongkun Zhang Mark Pollicott Carlangelo Liverani Sandro Vaienti Dmitry Dolgopyat 3:20-5:20 3:20-5:20 3:20-5:20 3:20-5:20 3:20-5:20 Mark Pollicott Sandro Vaienti Dmitry Dolgopyat Dmitry Dolgopyat Carlangelo Liverani

Monday Tuesday 8：30-11：00 8:30-11:00 Konstantin Khanin Mark Demers 2:00-4:30 2:00-4:30 Mark Demers Konstantin Khanin

Time | Speaker |
Title |

July 11-July 12 |
Huyi Hu
(Michigan State University) |
Introduction to transfer operator and decay of correlations Abstract: This lecture is to introduce some basic properties of transfer operators and some methods used in the study of decay of correlations. The topics include: - Definition and some properties of the transfer operators; - Quasi compactness and spectral gap of the operators; - Projective metrics; - Their applications for expanding maps; - Renew theory for polynomial decay; - Decay of correlations for hyperbolic systems: reduction to expanding systems and Banach spaces with distributions. |

July 11 , July 12 |
Jianyu Chen (Umass Amherst) |
Semiclassical analysis of the transfer/Koopman operator
Abstract: For some smooth dynamical systems, the semiclassical analysis provides a framework to convert the spectral property of the Koopman operator, the dual of the transfer operator, into pseudo-differential symbol calculus. The topics of this lecture include: - some basics on Sobolev spaces, pseudo-differential operators and Fourier integral operators; - Spectral radius estimate of the weighted Koopman operators; - Spectral gap and thus decay of correlations for expanding systems and its generic toral extension. |

July 13 ~ July 15 |
Mark Pollicott (University of Warwick) |
Thermodynamical formalism and Zeta functions Abstract: The most famous zeta function is the Riemann zeta function in number theory, In geometry the analogous complex function is the Selberg Zeta function. However, in order to analyze many of its properties in some important cases one needs to use thermodynmical formalism. Ideas of Ruelle are central to constructing the extension, exploting the connection with the transfer operator. We will also consider the important question of the location of the zeros. |

July 13 ~ July 15 |
Jose Ferreira Alves (University of Porto) |
Decay of correlations via towers for SRB measures of non-uniformly hyperbolic systems Abstract: The goal of this course is to deduce the existence and some statistical properties of SRB measures for non-uniformly hyperbolic dynamical systems. This will be achieved by mean of induced schemes and Young towers, and will be considered both for invertible and non-invertible discrete time dynamical systems. As applications we will consider certain classes of partially hyperbolic diffeomorphisms and non-uniformly expanding maps. |

July 14 ~ July 15 |
Nicolai Haydn (University of Southern Califonia) |
Entry and return times distributions |

July 18- July 19 | Richard Sharp (University of Warwick) |
Large deviations for dynamical systems Abstract:Large deviations theory is concerned with estimating the probability that averages are away from their limiting value. More precisely, let T be a dynamical system on X with an ergodic probability measure and let f be an observable on X. By the ergodic theorem, the Birkhoff averages of f converge to the expectation of f, almost surely. A large deviations result would estimate the probability that the averages are not close to the limit. More generally, one might study the probabilty of deviations of the average of the empirical distribution away from the invariant measure, in the space of probability measures, and hope for similar estimates. Results like this hold for hyperbolic systems and some systems with weaker hyperbolicity properties. We will discuss this and also the weaker polynomial estimates that hold in some other situations, for example intermittent systems. |

July 18 ~ July 19 |
Hongkun Zhang (Umass Amherst) |
Standard pairs and coupling methods in billiard systems Decay rates of correlations and other limiting theorem for hyperbolic systems with (or without) singularities are central questions in study statistical properties of these dynamical systems. Among various of methods, the tool of standard pairs and the coupling methods (first brought up by Nikolai Chernov and Dimtry Dolgopyat) are rather powerful in dealing with billiards and other systems with singularities. This minicourse will concentrate on the following topics: (1)Brief introduction of classical billiard; (2) Concenpts of standard pairs, standart families, coupling methods. |

July 19 ~ July 21 |
Sandro Vaienti (CNRS & Université de Provence) |
Sequential dynamical systems with respect to limit theorems and extreme value theory. Abstract: The course will cover the following topics: - Extreme Value Theory: position of the problem and application to dynamical systems; - Extreme value theory: the spectral approach. application to coupled map lattices; - Sequential Dynamical Systems: the expanding case with the almost sure invariance principle; - Sequential Dynamical Systems: the non-uniformly expanding case: loss of memory and central limit theorem. |

July 21 ~ July 22 |
Dmitry Dolgopyat (University of Maryland) |
Dynamics of Circle Rotations. The topics include: Ergodic sums of smooth functions with singularities, Ergodic sums of characteristic functions. Discrepancies, Poisson regime and their Application of the Poisson regime theorems to the ergodic sums of smooth functions with singularities, Poisson processes, and rates of recurrence. More information can be found in the notes at http://www.math.umd.edu/~dolgop/TorusRev15.pdf, Section 2, 3, 4, 5, 7.1, 7.2, 9.4. |

July 20 ~ July 22 |
Carlangelo Liverani (U. Roma Tor Vergata) |
Decay of correlations for some Anosov flows Title: Spectral analysis of transfer operators for smooth hyperbolic maps Abstract: This course will provide an introduction to the main ideas behind the construction of suitable Banach spaces for hyperbolic maps in a series of 3 lectures. 1) Recap of transfer operators and Banach spaces for expanding maps. Perturbation theory. Transfer operators for contracting maps: what to do? 2) Banach spaces for transfer operators for smooth hyperbolic maps. 3) Transfer operators with weights. A few comments about piecewise continuous maps. |

July 25 ~ July 26 |
Konstantin Khanin (University of Toronto) |
Random Lagrangian Systems The mini-course will look at a problem of action-minimizing trajectories for random Lagrangian Systems. The problem can be viewed as an extension of the Aubry-Mather theory to the case of random time-dependent Lagrangians. It is also connected with the theory of global solutions for random Hamilton-Jacobi equations. We shall describe results in the compact setting where the picture is rather complete at present. We also plan to discuss non-compact case where the situation is much less understood. There are many open questions there, some of them are related to the problem of KPZ (Kardar-Parisi-Zhang) universality. |

July 25 ~ July 26 |
Mark Demers (Fairfield University) |
Decay of correlations for some Anosov flows, cont’d Title: Spectral analysis of transfer operators for contact hyperbolic flows Abstract: This course will describe some essential features needed to adapt the analysis of discrete time transfer operators for maps to the semi-group of continuous time transfer operators for flows. 1) Adapting Banach spaces to hyperbolic flows; presence of a neutral direction and importance of the resolvent. 2) A spectral gap for the generator of the semi-group: an oscillatory integral and the contact form. A few comments about the extension to billiard flows. |