Reconstructing Semi-Markov Maps with Arbitrary Branch Monotonicity from Density Sequences

This paper presents a novel solution to the inverse Frobenius-Perron problem of reconstructing an unknown one-dimensional ergodic map from causal sequences of density functions generated by the map. The original solution to this problem successfully reconstructs members of the canonical map class (i.e., a subset of the piecewise linear semi-Markov maps), provided all of the map's branches are monotonically increasing. The map is reconstructed from a matrix estimate of its Frobenius-Perron operator, which governs the evolution of density functions under the map. The proposed solution first constructs a higher-order matrix estimate of this operator. A member of the newly proposed class of generalized hat maps, a superset of the canonical maps, is then constructed from this matrix estimate. The proposed solution successfully distinguishes between increasing and decreasing branches and enlarges the class of maps that can be reconstructed to canonical maps with any subset of decreasing branches. When used to reconstruct any piecewise linear semi-Markov map, the proposed solution generates a map with consistent invariant density and power spectrum characteristics, regardless of the unknown map's canonicity or branch monotonicity. Numerical examples that illustrate these characteristics of the proposed solution are presented.