Iterative Maps with a Given Ergodic Distribution

Abstract

The discovery that functions as simple as a quadratic curve could produce a chaotic and ergodic sequence of iterates as an iterative map has inspired many studies and surveys, on the behaviour of iterative functions and the sequence of iterates of these functions. One particularly well explored function is the logistic equation, the simplest of all non-linear curves, which held the interest of many mathematicians for a time. Aside from purely academic interest of how deterministic functions could lead to unpredictabilities, and the general curiosity of what surprises may lie behind the otherwise plain and simple functions, the attention of a few studies in particular had been on the equilibrium distribution over the sequence of itamendments that I coulderates of these iterative maps.

The Frobenius-Perron equation describes the relationship between the iterative maps and the distribution of the sequence of iterates, and is of interest particularly to people who wish to sample from a desired distribution, as the computation cost of performing these iterations are relatively low and convergence to the desired distribution is guaranteed, even in high dimensions. The direction of many studies had been on what distribution would be obtained by certain functions and how a class of functions would behave, however the aim of this thesis is the reverse.

A rearrangement of the Frobenius-Perron equation reveals that, though the invariant distribution is unique to the iterative map, the map itself is not unique to the invariant distribution; many iterative maps have the same invariant distribution. The aim of this study is to develop a method that backtracks from a desired distribution to construct an iterative map that has the desired equilibrium distribution, and hopefully generalize the method to all distributions, as well as to higher dimensions.