Abstract Bandtlow

Oscar Bandtlow: Lagrange-Chebyshev approximation of expanding maps
Abstract:
Thermodynamic formalism is a successful approach to study the long-term statistical behaviour of low-dimensional chaotic dynamical systems. A central object of this framework is the Ruelle transfer operator. Its spectral data, in particular, the leading eigenvalue and corresponding eigenvector, known as pressure and equilibrium measure in this context, can be used to obtain important dynamical and geometric invariants of the underlying system such as entropy, Lyapunov exponents, invariant measures, or Hausdorff dimension.

As there are very few examples for which spectral data of transfer operators can be computed exactly, it is of interest to look for numerical approximation schemes. One such method, known as the Ulam method, is applicable to rather general systems, but converges very slowly. Another method, known as the finite-section method, converges fast (exponentially, in fact, in certain cases) but is only applicable to analytic maps.

In this talk I will discuss a new and easily implementable method to approximate spectral data of transfer operators arising from piecewise expanding Markov maps of the interval, which, in a certain sense, `interpolates' between these two schemes. More precisely, as I will try to explain in the talk, this method adapts itself to the smoothness of the underlying map in the sense that the speed of convergence is

exponential if the map is analytic;
rapid (i.e. faster than algebraic) if the map is smooth;
algebraic if the map is C^k for k>2.