Ulam's method for Lasota-Yorke maps with holes

TL;DR

This paper adapts Ulam's method to Lasota-Yorke maps with holes, providing a convergent numerical scheme for escape rates, conditional invariant densities, and measures, even with large holes, supported by theoretical proofs and examples.

Contribution

It introduces a simple adaptation of Ulam's method for maps with holes, proving convergence of key dynamical quantities and enabling analysis of large holes.

Findings

Convergent sequences for escape rates and invariant densities are obtained.

The method applies to a broad class of maps, including Lorenz maps.

Numerical examples demonstrate the effectiveness of the approach.

Abstract

Ulam's method is a rigorous numerical scheme for approximating invariant densities of dynamical systems. The phase space is partitioned into connected sets and an inter-set transition matrix is computed from the dynamics; an approximate invariant density is read off as the leading left eigenvector of this matrix. When a hole in phase space is introduced, one instead searches for \emph{conditional} invariant densities and their associated escape rates. For Lasota-Yorke maps with holes we prove that a simple adaptation of the standard Ulam scheme provides convergent sequences of escape rates (from the leading eigenvalue), conditional invariant densities (from the corresponding left eigenvector), and quasi-conformal measures (from the corresponding right eigenvector). We also immediately obtain a convergent sequence for the invariant measure supported on the survivor set. Our approach allows us to consider relatively large holes. We illustrate the approach with several families of examples, including a class of Lorenz maps.