The exact rate of approximation in Ulam's method
Christopher Bose Rua Murray
This paper investigates the exact rate of convergence in Ulam's method: a well-known discretization scheme for approximating the invariant density of an absolutely continuous invariant probability measure for piecewise expanding interval maps. It is shown by example that the rate is no better than $O(\frac{\log n}{n})$, where $n$ is the number of cells in the discretization. The result is in agreement with upper estimates previously established in a number of general settings, and shows that the conjectured rate of $O(\frac{1}{n})$ cannot be obtained, even for extremely regular maps.
keywords: $L^1$ Error Bounds. Perron-Frobenius Operator Interval Maps Approximation of Absolutely Continuous Invariant Measures
Quasi-invariant measures, escape rates and the effect of the hole
Wael Bahsoun Christopher Bose
Let $T$ be a piecewise expanding interval map and $T_H$ be an abstract perturbation of $T$ into an interval map with a hole. Given a number , 0 < < l, we compute an upper-bound on the size of a hole needed for the existence of an absolutely continuous conditionally invariant measure (accim) with escape rate not greater than -ln(1-). The two main ingredients of our approach are Ulam's method and an abstract perturbation result of Keller and Liverani.
keywords: Transfer Operator Interval Maps Ulam's Method. Escape Rates
Minimum 'energy' approximations of invariant measures for nonsingular transformations
Christopher Bose Rua Murray
We study variational methods for rigorous approximation of invariant densities for a nonsingular map $T$ on a Borel measure space. The general method takes the form of a convergent sequence of optimization problems on $L^p$, $1 \leq p < \infty$ with a convex objective and finite moment constraints. Provided $T$ admits an invariant density in the appropriate $L^p$ space, weak convergence of the sequence of optimal solutions is observed; norm convergence can be obtained when the objective is a Kadec functional. No regularity or expansiveness assumptions on $T$ need to be made, and the method applies to maps on multidimensional domains. Objectives leading to norm convergence include Entropy, 'Energy' and 'Positively Constrained Energy'.
   Explicit solutions for the finite moment problems in the case of the 'Energy' functional are derived using duality - the optimality condition is then a linear algebra problem. Strong duality is obtained even though the dual functional may not be coercive and the set of moment test functions is not assumed to be pseudo-Haar. Finally, some numerical studies are presented for the case of moment test functions derived from a finite partition of the dynamical phase space and the results are compared with Ulam's method.
keywords: Frobenius-Perron operator entropy normal convex integrand duality. Kadec moment problem Invariant measure
Deterministic representation for position dependent random maps
Wael Bahsoun Christopher Bose Anthony Quas
We give a deterministic representation for position dependent random maps and describe the structure of its set of invariant measures. Our construction generalizes the skew product representation of random maps with constant probabilities. In particular, we establish one-to-one correspondence between eigenfunctions corresponding to eigenvalues of unit modulus for the Frobenius-Perron (transfer) operator of the random map and for those of the skew. An immediate consequence is one-to-one correspondence between absolutely continuous invariant measures (acims) for the position dependent random map and acims for its deterministic representation.
keywords: Random map skew product absolutely continuous invariant measure.

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