Set-wise methods for the invariant measure problem: Convergence?
In 1960 Ulam proposed discretising the Perron-Frobenius operator for a non-singular 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 Frobenius-Perron 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 Frobenius-Perron 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 set-oriented 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.