Ulam's method for Lasota-Yorke maps with holes
This provides a rigorous numerical framework for computing escape statistics in chaotic systems with holes, benefiting researchers in dynamical systems and ergodic theory.
The authors prove that Ulam's method, adapted for Lasota-Yorke maps with holes, yields convergent sequences for escape rates, conditional invariant densities, quasi-conformal measures, and invariant measures on the survivor set, even for relatively large holes. Numerical examples, including Lorenz maps, illustrate the approach.
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.