Ulam's method for Lasota-Yorke maps with holes

Bose, Christopher, Froyland, Gary, Gonzalez-Tokman, Cecilia and Murray, Rua (2014) Ulam's method for Lasota-Yorke maps with holes. SIAM Journal on Applied Dynamical Systems, 13 2: 1010-1032. doi:10.1137/130917533

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Author Bose, Christopher
Froyland, Gary
Gonzalez-Tokman, Cecilia
Murray, Rua
Title Ulam's method for Lasota-Yorke maps with holes
Journal name SIAM Journal on Applied Dynamical Systems   Check publisher's open access policy
ISSN 1536-0040
Publication date 2014
Year available 2014
Sub-type Article (original research)
DOI 10.1137/130917533
Open Access Status File (Publisher version)
Volume 13
Issue 2
Start page 1010
End page 1032
Total pages 23
Place of publication Philadelphia, PA, United States
Publisher Society for Industrial and Applied Mathematics
Collection year 2014
Language eng
Abstract 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.
Keyword Lasota-Yorke maps
Open dynamical systems
Ulam's method
Q-Index Code C1
Q-Index Status Provisional Code
Institutional Status Non-UQ

Document type: Journal Article
Sub-type: Article (original research)
Collections: School of Mathematics and Physics
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Created: Wed, 29 Apr 2015, 15:44:31 EST by Kay Mackie on behalf of School of Mathematics & Physics