Seminar 4.30.16 Misiurewicz

Speaker: Michal Misiurewicz (IUPUI)

Title: Entropy locking

Abstract: Consider discontinuous piecewise linear interval maps with two pieces, where the map is increasing on one piece and decreasing on the other piece. Often the topological entropy depends only on the slopes, not on the size of the jump at the discontinuity point. We present a simple explanation of this phenomenon. This is joint work with David Cosper.

Seminar 4.7.16 Webb

Speaker: Ben Webb (Brigham Young University)

Title: Intrinsic stability of time-delayed networks and multidimensional dynamical systems

Abstract: In real networks the time it takes to send and process information inevitably leads to time delays in the network’s dynamics. These time-delays are important to the network’s dynamics as they are often the source of instability and poor performance. In fact, time delays can both destabilize stable systems and stabilize unstable systems, depending on the system and where these delays are placed. In this talk we introduce a stronger notion of stability that is preserved under changes to a network’s structure of delays. This we call intrinsic stability, which can be used to simplify the stability analysis for both dynamical networks as well as multidimensional systems. This work is joint with L. A. Bunimovich.

Seminar 3.24.16 Moreira

Speaker: Joel Moreira (Ohio State)

Title: Non-linear monochromatic patterns in N via topological dynamics

Abstract: Since Furstenberg’s seminal paper in 1977 providing an ergodic theoretic proof of Szemeredi’s theorem on arithmetic progressions, dynamical systems methods have been a very successful tool in obtaining combinatorial results. A central problem in Ramsey theory is to understand and classify which polynomial patterns can be found “monochromatically” in any arbitrary finite coloring of the natural numbers. In particular, the question of whether any finite coloring of the natural numbers yields a monochromatic pattern of the form {x+y,xy} has remained unanswered for several years. In this talk I will investigate dynamical approaches to this and related questions, employing techniques from topological dynamics.

Seminar 3.17.16 Fraser

Speaker: Jon Fraser (Manchester)

Title: Inhomogeneous iterated function systems

Abstract: Inhomogeneous iterated function systems are natural generalisations of the classic iterated function systems, commonly used to generate examples of fractal sets. The key difference is that one begins with a fixed ”condensation” set which is then dragged into the construction by the iterates of mappings in the IFS. Such systems have applications in image compression in situations where one wants to produce an image with lots of similar objects appearing at different scales, like a flock of seagulls or a forest. I will review some structural properties of the attractors of such systems and go on to discuss their dimension theory. Some of this talk will be joint work with Simon Baker (Reading) and Andras Mathe (Warwick).

Seminar 2.25.16 Bunimovich

Speaker: Leonid Bunimovich (Georgia Institute of Technology)

Title: Finite time properties of transport in chaotic systems.

Abstract: We are used to the idea that only asymptotic in time properties of systems with “complex” dynamics should (and could) be understood rigorously. Therefore basically always asymptotics when time goes to infinity or integration over an infinite time interval are involved. I will discuss  finite time properties of survival, first passage and recurrence probabilities for some classes of dynamical and stochastic systems.

Seminar 2.18.16 Shah

Speaker: Nimish Shah (Ohio State)

Title: Equidistribution of stretching translates of curves on homogeneous spaces

Abstract: We consider a finite piece C of an analytic curve on a minimal expanding (abelian) horospherical subgroup of G=SL(n,R) associated to a certain diagonal element g in G. We consider the subgroup action of G on a finte volume homogeneous space X, and consider the trajectory of C from some point x in X. We want to understand algebraic conditions on C which ensure that in the limit, the translates of the curve Cx by powers of g get equidistributed in the (homogeneous) closure of the G-orbit of x. In this talk we describe some recent joint work with Lei Yang on this problem.

Such results have applications to metric properties of Diophantine approximation- namely, to show non-improvability of Dirichlet’s approximation on curves.

Seminar 2.11.16 Todd

Speaker: Mike Todd (St. Andrews)

Title: Continuity of measures

Abstract: Given a convergent family of interval maps and the associated family of SRB measures, one might hope that the measures would converge to the SRB measure of the limit map.  In non-uniformly hyperbolic systems, this naive approach can fail. I’ll give sharp conditions on precisely when this failure occurs for a very general class of maps. This is part of a wider study of continuity of thermodynamic quantities in collaboration with Neil Dobbs.

Seminars for Spring 2016

Currently our schedule for the Spring is as follows.

Feb 11: Mike Todd (St. Andrews)

Feb 18: Nimish Shah (Ohio State)

Feb 25: Leonid Bunimovich (Georgia Tech)

Mar 17:  Jon Fraser (Manchester)

Mar 24: Joel Moreira (Ohio State)

Apr 7: Ben Webb (Brigham Young)

Apr 21: Kostya Medynets (United States Naval Academy)

Apr 30: Michal Misiurewicz (IUPUI)

May 12: COLLOQUIUM: Mark Pollicott (Warwick)