# American Institute of Mathematical Sciences

July  1998, 4(3): 535-557. doi: 10.3934/dcds.1998.4.535

## Approximation error for invariant density calculations

 1 Centre for Nonlinear Dynamics, University College London, Gower St, London WC1E 6BT, United Kingdom

Received  May 1997 Revised  March 1998 Published  April 1998

Let $T:[0,1]\rightarrow[0,1]$ be an expanding piecewise-onto map with bounded distortion and countably many intervals of monotonicity. We prove a uniform bound on the rate of exponential convergence to equilibrium for iterates of the Perron-Frobenius operator. The quantitative information thus obtained is applied to prove explicit error bounds for Ulam's method for approximating invariant measures. The approach also yields rates of mixing for the matrix representations of Ulam approximation. "Monte-Carlo" type simulations of the scheme are discussed. The method of proof is applicable to multi-dimensional transformations, although the only generalisations presented here are to a limited class of "non-onto" one-dimensional transformations.
Citation: Rua Murray. Approximation error for invariant density calculations. Discrete and Continuous Dynamical Systems, 1998, 4 (3) : 535-557. doi: 10.3934/dcds.1998.4.535
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