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 2014-01-01
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
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
Non HERDC
 
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Created: Thu, 30 Apr 2015, 01:44:31 EST by Kay Mackie on behalf of School of Mathematics & Physics