# Spectral Galerkin methods for transfer operators in uniformly expanding dynamics

## TL;DR

This paper introduces a spectral Galerkin method for transfer operators in uniformly expanding maps, providing highly accurate statistical estimates efficiently and with rigorous validation, advancing computational tools for chaotic dynamics analysis.

## Contribution

The paper develops a spectral Galerkin approach with proven convergence rates and introduces two algorithms, including a rigorously validated one, for efficient and accurate statistical analysis of chaotic systems.

## Key findings

- Exponential accuracy in statistical estimates achieved with polynomial computational effort.
- Rigorous bounds on statistical quantities surpass previous methods.
- Fast adaptive algorithm produces double-precision estimates in seconds.

## Abstract

Markov expanding maps, a class of simple chaotic systems, are commonly used as models for chaotic dynamics, but existing numerical methods to study long-time statistical properties such as invariant measures have a poor trade-off between computational effort and accuracy. We develop a spectral Galerkin method for these maps' transfer operators, estimating statistical quantities using finite submatrices of the transfer operators' infinite Fourier or Chebyshev basis coefficient matrices. Rates of convergence of these estimates are obtained via quantitative bounds on the full transfer operator matrix entries; we find the method furnishes up to exponentially accurate estimates of statistical properties in only a polynomially large computational time. To implement these results we suggest and demonstrate two algorithms: a rigorously-validated algorithm, and a fast, more convenient adaptive algorithm. Using the first algorithm we prove rigorous bounds on some exemplar quantities that are substantially more accurate than previous. We show that the adaptive algorithm can produce double floating-point accuracy estimates in a fraction of a second on a personal computer.