Mathematics – Dynamical Systems
Scientific paper
2012-04-11
Mathematics
Dynamical Systems
22 pages, 6 figures
Scientific paper
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 is non-perturbative and allows us to consider relatively large holes.
Bose Christopher
Froyland Gary
González-Tokman Cecilia
Murray Rua
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