Explore fundamental concepts and advanced techniques in dynamical systems through this comprehensive summer school program from ICTP Mathematics. Master basic dynamics concepts including rotations of the circle, doubling maps, Gauss maps and continued fractions, while developing understanding of symbolic codings and invariant measures. Delve into structural stability and renormalization through practical examples and theoretical frameworks. Study ergodic theory in smooth dynamical systems, beginning with linear examples of rotations and doubling maps on the circle, then advancing to nonlinear smooth systems. Learn essential tools for establishing ergodicity including distortion estimates, density points, invariant foliations and absolute continuity. Focus on the ergodic theory of Anosov diffeomorphisms as important models of chaotic dynamical systems. Investigate renormalization techniques in entropy zero systems, examining how the Gauss map and continued fractions serve as tools to renormalize rotations across multiple scales. Discover the characterization of Sturmian sequences arising from symbolic coding of rotation trajectories and their connections to cutting sequences for billiards. Study interval exchange maps (IETs) as generalizations of rotations, utilizing the Rauzy-Veech algorithm for renormalization. Apply these concepts to analyze invariant measures, unique ergodicity, and deviations of ergodic averages for IETs. Participate in regular tutorial and exercise sessions that form an essential component of the learning experience, with dedicated support for women in mathematics through panel discussions and mentoring opportunities.

Syllabus

Rotations of the circle and renormalization 1
Piecewise expanding maps of the circle
Introduction to ergodic theory 1
Rotations of the circle and renormalization 2
Piecewise expanding maps of the circle 2
Introduction to ergodic theory 2
Rotations of the circle and renormalization 3
Introduction to ergodic theory 3
Piecewise expanding maps of the circle 3
Piecewise expanding maps of the circle 4
Rotations of the circle and renormalization 4
Introduction to ergodic theory 4
Rotations of the circle and renormalization 5
Introduction to ergodic theory 5
Piecewise expanding maps of the circle 5
Ergodicity in smooth dynamics 1
Ergodicity in smooth dynamics 2
Renormalization in entropy zero systems 1
Renormalization in entropy zero systems 2
Ergodicity in smooth dynamics 3
Ergodicity in smooth dynamics 4
Ergodicity in smooth dynamics 5
Renormalization in entropy zero systems 3
Ergodicity in smooth dynamics 6
Ergodicity in smooth dynamics 7
Renormalization in entropy zero systems 4
Ergodicity in smooth dynamics 8
Renormalization in entropy zero systems 5
Ergodicity in smooth dynamics 9
Ergodicity in smooth dynamics 10