Sydney Dynamics Group Seminar: Nicholas Fleming, University of New South Wales -- Sydney Dynamics Group Seminar, Nicholas Fleming (UNSW), Statistical properties of certain 2D mostly expanding fast-slow systems

Seminar : Sydney Dynamics group Seminar.  
Day/time : Thursday, October 22 at 4:00 PM
Location : SMRI seminar room (A12 Macleay Room 301).  
Speaker : Nicholas Fleming, University of New South Wales.  

Title: Statistical properties of certain 2D mostly expanding fast-slow systems 

Abstract: 

This is joint work with Jacopo De Simoi and Kasun Fernando.  We consider a class of
fast-slow systems on the 2-torus, where the fast variable is given by a family of
sufficiently smooth expanding maps.  Such systems obey an averaging principle: at an
appropriate time-scale the slow variable is approximated by an averaged ODE.  De Simoi
and Liverani initiated a programme to study the behaviour of these systems at
arbitrarily long time-scales by considering their statistical properties (existence of
physical measures and the rate of decay of correlations).  Under the assumptions that
the averaged ODE has exactly one sink and that both Lyapunov exponents of the system are
positive, we prove that there is a unique physical (SRB) measure.  Moreover, we
establish exponential decay of correlations for Hölder observables, with explicit,
nearly optimal bounds on the exponential decay rate.  Our result provides a ‘mostly
expanding’ counterpart to the work of De Simoi–Liverani, who treated such systems in
the ‘mostly contracting’ case (i.e.  when the system has one negative Lyapunov
exponent).