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Mathematics of Computation

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Computing the invariant measure and the Lyapunov exponent for one-dimensional maps using a measure-preserving polynomial basis


Authors: Philip J. Aston and Oliver Junge
Journal: Math. Comp. 83 (2014), 1869-1902
MSC (2010): Primary 37M25, 65P20
DOI: https://doi.org/10.1090/S0025-5718-2013-02811-6
Published electronically: November 12, 2013
MathSciNet review: 3194133
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Abstract: We consider a generalisation of Ulam's method for approximating invariant densities of one-dimensional maps. Rather than use piecewise constant polynomials to approximate the density, we use polynomials of degree $ n$ which are defined by the requirement that they preserve the measure on $ n+1$ neighbouring subintervals. Over the whole interval, this results in a discontinuous piecewise polynomial approximation to the density. We prove error results where this approach is used to approximate smooth densities. We also consider the computation of the Lyapunov exponent using the polynomial density and show that the order of convergence is one order better than for the density itself. Together with using cubic polynomials in the density approximation, this yields a very efficient method for computing highly accurate estimates of the Lyapunov exponent. We illustrate the theoretical findings with some examples.


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Additional Information

Philip J. Aston
Affiliation: Department of Mathematics, University of Surrey, Guildford, Surrey GU2 7XH, United Kingdom
Email: P.Aston@surrey.ac.uk

Oliver Junge
Affiliation: Technische Universität München, Zentrum Mathematik, Boltzmannstr. 3, D-85747 Garching, Germany
Email: oj@tum.de

DOI: https://doi.org/10.1090/S0025-5718-2013-02811-6
Received by editor(s): November 23, 2011
Received by editor(s) in revised form: September 17, 2012
Published electronically: November 12, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.