The exact rate of approximation in Ulam's method

Christopher Bose; Rua Murray

doi:10.3934/dcds.2001.7.219

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2001, Volume 7, Issue 1: 219-235. Doi: 10.3934/dcds.2001.7.219

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The exact rate of approximation in Ulam's method

Christopher Bose^1, and
Rua Murray^2,

1.
Department of Mathematics and Statistics, University of Victoria, P.O. Box 3045, Victoria, BC, Canada V8W 3P4
2.
Department of Mathematics, University of Waikato, Private Bag 3105, Hamilton

Received: June 30, 2000

Revised: October 31, 2000

Published: December 2000

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Abstract

This paper investigates the exact rate of convergence in Ulam's method: a well-known discretization scheme for approximating the invariant density of an absolutely continuous invariant probability measure for piecewise expanding interval maps. It is shown by example that the rate is no better than $O(\frac{\log n}{n})$, where $n$ is the number of cells in the discretization. The result is in agreement with upper estimates previously established in a number of general settings, and shows that the conjectured rate of $O(\frac{1}{n})$ cannot be obtained, even for extremely regular maps.

Keywords:

Interval Maps,
Approximation of Absolutely Continuous Invariant Measures,
Perron-Frobenius Operator,
$L^1$ Error Bounds.

Mathematics Subject Classification: 28D05, 41A25, 41A44.

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