ICERM Semester Program on "Dimension and Dynamics"
(February 1 - May 6, 2016)
Registration opens winter 2014
- Dmitry Dolgopyat
(University of Maryland)
- Mike Hochman
(Hebrew University of Jerusalem)
- Izabella Laba
(University of British Columbia)
- Stefano Luzzatto
- Yakov Pesin
(Penn State University)
- Mark Pollicott
- Joerg Schmeling
- Boris Solomyak
(University of Washington)
- Warwick Tucker
[Photo Credit: Mark A. Wilson]
Since its introduction by Felix Hausdorff in 1919, the concept of the Hausdorff dimension of sets and measures has been a versatile and powerful tool in classical analysis, geometry and geometric measure theory, mathematical physics and their numerous applications. However, there has been a particularly important symbiosis between dynamical systems and dimension theory. This connection arises both from application of dimension theory to the classification and geometric analysis of dynamical systems (and their invariant sets and measures), and the fact that many classical objects of study in mathematics arise from (sometimes implicit) dynamical systems, which often play a role in the dimension theory of said objects.
Recently, there has been substantial progress on a number of central problems in dimension theory, and while many old problems remain, many new ones have also presented themselves. These include a deeper understanding of the relationship between dimension, entropy and Lyapunov exponents; the recent strengthenings of the Marstrand projection theorem and its implications for dimensions of sums of Cantor sets and the connections with number theory; multifractal analysis of ergodic averages, particularly the recent advances for multiple ergodic averages; and improved computational methods for obtaining effective bounds on Hausdorff dimension
This proposed program aims to set the stage for further progress on the many open problems in the field.
It is a challenging classical problem to compute numerically dimensions of given fractals. In recent years, computer-aided proofs in mathematical analysis have gained an increasing presence in mathematical research. One reason for this field's growing impact is due to its ability to produce high-quality quantitative information about global, nonlinear problems. As a consequence, solutions to a large class of previously intractable problems are now within reach and recently several long-standing conjectures have been verified using rigorous computations
During the semester we will focus on three specific aspects of the interaction between these two areas: (i) Ergodic, algebraic and combinatorial methods in dimension theory (ii) Computations in fractal geometry in dynamical systems; and (iii) Fractal geometry and hyperbolic dynamics.