Finite Element Maximum Entropy Method for Approximating Absolutely Continuous Invariant Measures

In a chaotic dynamical system, the eventual behavior of iterates of initial points of a map is unpredictable even though the map is deterministic. A system which is chaotic in a deterministic point of view may be regular in a statistical viewpoint. The statistical viewpoint requires the study of absolutely continuous invariant measure (ACIM) of a map with respect to the Lebesgue measure. An invariant density of the Frobenius-Perron (F-P) operator associated with a nonsingular map is employed to evaluate an ACIM of the map. The ACIM is a key factor for studying the eventual behavior of iterates of almost all initial points of the map. It is difficult to obtain an invariant density of the F-P operator in an exact mathematical form except for some simple maps. Different numerical schemes have been developed to approximate such densities. The maximum entropy principle gives a criterion to select a least-biased density among all densities satisfying a system of moment equations. In this principle, a least-biased density maximizes the Boltzmann entropy. In this dissertation, piecewise quadratic functions and quadratic splines are used in the maximum entropy method to calculate the L1 errors between the exact and the approximate invariant densities of the F-P operator associated with nonsingular maps defined from [0;1] to itself. The numerical results are supported by rigorous mathematical proofs. The L1 errors between the exact and approximate invariant densities of the Markov operator associated with Markov type position dependent random maps, defined from [0;1] to itself, are calculated by using the piecewise linear polynomials maximum entropy method

Invariant measures for random maps via interpolation

Let T = {τ1, τ2, ..., τK; p1, p2, ..., pK} be a position dependent random map on [0, 1], where {τ1, τ2, ..., τK} is a collection of nonsingular maps on [0, 1] into [0, 1] and {p1, p2, ..., pK} is a collection of position dependent probabilities on [0, 1]. We assume that the random map T has a unique absolutely continuous invariant measure μ with density f*. Based on interpolation, a piecewise linear approximation method for f* is developed and a proof of convergence of the piecewise linear method is presented. A numerical example for a position dependent random map is presented