Ulam's Method for Lasota- Yorke Maps with Holes

作者：Bose Christopher^*; Froyland Gary; Gonzalez Tokman Cecilia; Murray Rua

来源：SIAM Journal on Applied Dynamical Systems, 2014, 13(2): 1010-1032.

DOI：10.1137/130917533

• open dynamical systems
• Ulam's method
• Lasota-Yorke maps

Ulam's method is a rigorous numerical scheme for approximating invariant densities of dynamical systems. The phase space is partitioned into a grid of connected sets, and a set-to-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 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-like maps.

• 出版日期2014