Abstract

In this PhD thesis we are concerned with the existence of the invariant measures and the absolutely continuous invariant measures under one-dimensional transformations. The thesis consists of two articles.
We establish the existence of invariant measures for random maps with constant probabilities and for nonautonomous random dynamical systems generalizing Krylov-Bogoliubov Theorem. We present results on the existence of an absolutely continuous invariant measure for the nonautonomous random maps on [a,b] using the theory of bounded variation.

We study the dynamics of a new family of transformations. We defined a general formula for the density function for any transformation belonging to our family, and we find some special properties for this family. This allowed us to study the random maps with constant probabilities based on these maps and to prove that the density function f of random map which is constructed from our family maps T= { τ1, τ2, ..., τn; p1, p2, ..., pn} is the combination f = p1f1+ p2f2+ ...+ pnfn, where f1, f2, ..., fn are a density functions of τ1, τ2, ..., τn respectively. We defined another family of transformations, and we proved that the density functions for any transformations belonging to this family are f(x)=1. We present an example to find the density function of the random maps by conjugations.

We created two classes of chaotic maps with desired invariant densities using two methods of solving the inverse Frobenius-Perron problem (IFPP). We studied the Lyapunov exponent and the autocorrelation properties for one of these classes.