Spectral Galerkin methods for transfer operators in uniformly expanding dynamics

Abstract

We present spectral methods for numerically estimating statistical properties of uniformly-expanding Markov maps. We prove bounds on entries of the Fourier and Chebyshev basis coefficient matrices of transfer operators, and show that as a result statistical properties estimated using finite-dimensional restrictions of these matrices converge at classical spectral rates: exponentially for analytic maps, and polynomially for multiply differentiable maps.Our proof suggests two algorithms for the numerical computational statistical properties of uniformly expanding Markov maps: a rigorouslyvalidated algorithm, and a fast adaptive algorithm. We give illustrative results from these algorithms, demonstrating that the adaptive algorithm produces estimates of many statistical properties accurate to 14 decimal places in less than one-tenth of a second on a personal computer.

Accurate bounds on Lyapunov exponents for expanding maps of the interval

“…We obtain the following values: which are each presented to 130 significant figures. In particular, with these choices Theorem 1.4 yeilds that ε This value has previously been computed by Wormell [20] and her result agrees with the above. In the present approach, the simplicity of the functions p and q is the source of the efficiency of the approach.…”

“…It is therefore useful to have a rigorous and effective estimate of these values, in particular, in the setting of one dimensional expanding maps for an absolutely continuous invariant probability measure. This problem has attracted the attention of many authors who have employed a variety of different methods (see [5], [12], [20]).…”

“…Another approach is based on periodic points method [10]. Recent work by Wormell [20] is based on the Galerkin spectral method originally developed for PDEs. In this note we present an alternative approach, which starts with the spectral Chebyshev collocation method, also initially developed for PDEs [6, §3].…”

Rigorous Computation of Linear Response for Intermittent Maps

“…(1) Interval Arithmetics and rigorous contractors as the Interval Newton Method and the Shooting Method [42] (2) discretization of the transfer operator, using the Ulam and Chebyshev basis [20,19,43] (3) a priori estimate on the tail of a series and rigorous bounds for a finite number of terms.…”

Efficient computation of statistical properties of intermittent dynamics

“…The induced map is uniformly expanding (see Figure 2), and it is therefore possible to apply results on uniformly expanding dynamics to it, as well as various numerical methods [18,5,4]. The non-mixing dynamics near the fixed point poses a problem for obtaining accurate numerical estimates for these maps.…”