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January  2001, 7(1): 219-235. doi: 10.3934/dcds.2001.7.219

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, Canada
2. 

Department of Mathematics, University of Waikato, Private Bag 3105, Hamilton, New Zealand

Received  July 2000 Revised  November 2000 Published  November 2000

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: $L^1$ Error Bounds., Perron-Frobenius Operator, Interval Maps, Approximation of Absolutely Continuous Invariant Measures.
Mathematics Subject Classification: 28D05, 41A25, 41A4.
Citation: Christopher Bose, Rua Murray. The exact rate of approximation in Ulam's method. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 219-235. doi: 10.3934/dcds.2001.7.219
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