Abstract

This thesis delves into three areas of research on dynamical systems. First, it explores the existence and exactness of Absolutely Continuous Invariant Measures (ACIM) for piecewise convex maps with countable (infinite) number of branches. Second, it employs Ulam’s method to approximate the density function of these ACIMs. Third, it investigates the existence of Absolutely Continuous Invariant Measures for piecewise concave maps using the technique of conjugation. For the first topic, we examine the existence and uniqueness of ACIMs within two distinct classes, denoted as T ∞
pc (I) and T ∞,0 pc (I), which together encompass piecewise convex maps τ : I =[0, 1] → [0, 1] with countable number of branches. We establish the necessary conditions under which these maps possess a unique ACIM, presenting multiple illustrative examples of ACIM existence.
Our findings are based on the analysis of the Frobenius-Perron operator associated with these maps, utilizing analytical techniques to gain insights into the Frobenius-Perron operator’s properties.
The main purpose of the second part of this thesis is to approximate τ by the map τn, where we construct a sequence τn with a finite number of branches. Then, approximate τn by Ulam’s method. Since piecewise convex maps have countable (infinite) number of branches, the convergence of Ulam’s method becomes more challenging, and complexity makes it harder to find a suitable sequence of approximating functions that can accurately analyze the behavior of this system across all branches.
The primary contribution of this Ph.D. thesis lies in the generalization of the existence of absolutely continuous invariant measures for piecewise convex maps defined on an interval with an infinite number of branches. In the case of T ∞pc (I), we examine piecewise convex maps with an infinite number of branches and arbitrary countable number of limit points for partition points separated from 0. For T ∞,0pc (I), we consider piecewise convex maps with countable number of branches and partition points that converge to 0. Throughout the thesis, we investigate Absolutely Continuous Invariant Measures (ACIM) for τ ∈ T ∞pc (I) and τ ∈ T ∞,0pc (I), along with exploring non-autonomous dynamical systems of maps within these classes and scrutinize the existence of ACIMs for their limit maps.
Furthermore, we investigate the approximation for ACIMs associated with piecewise convex maps with an infinite number of branches by employing Ulam’s method. This computational approach is a practical way to estimate the density functions of ACIMs and thereby facilitate their numerical analysis. We then extended our research area on ACIM for piecewise concave maps with countable number of branches. We examine the existence and uniqueness of ACIMs for two distinct classes, T ∞pcv(I) and T ∞,1pcv (I), which encompass piecewise concave mappings denoted as σ. We utilize the concept of conjugation with piecewise convex maps τ to demonstrate that σ conserves a normalized absolutely continuous invariant measure with a density that exhibits increasing behavior.