Setwise methods for the invariant measure problem: Convergence? (2012)
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Type of Content
Conference Contributions  OtherPublisher
University of Canterbury. Mathematics and StatisticsCollections
Abstract
In 1960 Ulam proposed discretising the PerronFrobenius operator for a nonsingular map (T;X) by projecting L1(X) onto the subspace of piecewise constant functions with respect to a fixed partition of subsets of X. Ulam's conjecture was that as the partition is refined, the fixed points of the approximation scheme should converge in L1 to a fixed point of the FrobeniusPerron operator. Thus "Ulam's method" was born! Li (1976) proved the conjecture for piecewise C2 expanding interval maps, and further results have been obtained by many authors over the subsequent decades. It is now clear that most of these results rely on strong analytical control of the spectrum of the FrobeniusPerron operator on suitable Banach spaces embedded in L1. Indeed, in such settings, useful convergence rates can be obtained (for example, by using the spectral perturbation machinery of Keller and Liverani). However, applying these results to new classes of maps can be extremely diffcult (or impossible); this is especially true examples coming from real applications. In this sense, a satisfactory proof of Ulam's conjecture remains elusive. This talk will survey the ideas above, and describe a variational framework in which Ulam's method arises as one possible approximation scheme (joint work with C Bose). Analytical proofs of convergence can come "cheaply" in a variety of settings where the spectral perturbation approach does not apply, including some open systems. In keeping with the setoriented theme of the conference, implementation relies on being able to compute intersections of elements of a partition of X, and topological features of the intersections turn out to be of great importance for the feasibility of the methods.
Citation
Murray, R. (2012) Setwise methods for the invariant measure problem: Convergence?. Sydney, Australia: International workshop on Set Oriented Numerics (SON2012), 36 Sep 2012.This citation is automatically generated and may be unreliable. Use as a guide only.
ANZSRC Fields of Research
49  Mathematical sciences::4904  Pure mathematics::490408  Operator algebras and functional analysis49  Mathematical sciences::4904  Pure mathematics::490409  Ordinary differential equations, difference equations and dynamical systems
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