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September  2010, 26(3): 1007-1018. doi: 10.3934/dcds.2010.26.1007

Ulam's method for some non-uniformly expanding maps

1. 

Department of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand

Received  March 2009 Revised  September 2009 Published  December 2009

Certain dynamical systems on the interval with indifferent fixed points admit invariant probability measures which are absolutely continuous with respect to Lebesgue measure. These maps are often used as a model of intermittent dynamics, and they exhibit sub-exponential decay of correlations (due to the absence of a spectral gap in the underlying transfer operator). This paper concerns a class of these maps which are expanding (with convex branches), but admit an indifferent fixed point with tangency of $O(x^{1+\alpha})$ at $x=0$ ($0<\alpha<1$). The main results show that invariant probability measures can be rigorously approximated by a finite calculation. More precisely: Ulam's method (a sequence of computable finite rank approximations to the transfer operator) exhibits $L^1$ - convergence; and the $n$th approximate invariant density is accurate to at least $O(n^{-(1-\alpha)^2})$. Explicitly given non-uniform Ulam methods can improve this rate to $O(n^{-(1-\alpha)})$.
Citation: Rua Murray. Ulam's method for some non-uniformly expanding maps. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 1007-1018. doi: 10.3934/dcds.2010.26.1007
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