Martingale Techniques and Anisotropic Banach Spaces
August 7 – 18, 2017
The  main  objective  is to  explore the  applicability  of martingale techniques  in the setting  of invertible  systems  using the  recently developed ‘adapted’  Banach  spaces pioneered by Liverani, Baladi, Gouëzel, Tsujii, Keller, Demers and others.

The application  of martingale  theory  to the study  of statistical properties of dy- namical  systems  was pioneered  by M. Gordin  in 1969.  By studying the  action  of the  transfer  operator  (the  dual  to the  action  of composing an observable  with  the transformation) the Gordin  approach  expresses an observable on the  system as the sum of a reverse martingale  difference and a coboundary.

Recent work has investigated the action of the transfer operator directly on certain anisotropic  Banach  spaces.  This  approach  avoid introducing symbolic coding, and allows perturbation of the system.  However, the martingale  technique  of Gordin has not  yet  been adapted to  this  context although  in certain  situations  it  enables  one to prove stronger  results. Examples  include convergence of stochastic  integrals  and rates of convergence in the weak invariance principle, where the primary existing probabilistic  tools are via martingale  theory.   The  collaboration  will be to develop ways of modifying Gordin’s approach  to the setting  of anisotropic Banach spaces.


Mark Demers (Fairfield University)
Ian Melbourne (University of Warwick)
Matthew Nicol (University of Houston)