ULAM'S METHOD FOR SOME NON-UNIFORMLY EXPANDING MAPS

作者：Murray Rua^*

来源：Discrete and Continuous Dynamical Systems, 2010, 26(3): 1007-1018.

DOI：10.3934/dcds.2010.26.1007

• indifferent fixed point
• invariant measure approximation
• non-uniformly expanding dynamical system
• mixing time
• polynomial decay of correlations
• Ulam's method

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 nth 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))).

• 出版日期2010-3