Research Projects

This proposal explores the dynamics arising from Γ−actions on a topological group X, where Γ is a subgroup of Aut(X). A pair (X, Γ) is considered a dynamical system if γ is an automorphism of X and Γ = {γ^n: n ∈ Z}. The ergodic theorems form an important part of the study on ergodicity, with the initial theorems being due to Birkhoff and von Neumann. The space average and pointwise average are two concepts related to Z−actions. The space average can be modified as (1/|D|)∑_(γ∈D) f(γ.x), where D is a finite subset of Γ. However, for convergence, D is not generally allowed to vary over all finite subsets of Γ; instead, a collection of finite subsets of Γ is properly chosen and D is allowed to vary over only this collection. The ratio ergodic theorem is another type of theorem introduced by Hopf. Although these theorems and the ratio ergodic theorem are well studied, they are not considered for general dynamical systems. The notion of convergence and the collection of finite subsets may vary from one point of view to another. The proposal aims to find a large enough collection of finite subsets of Γ for a dynamical system (X, Γ) for which the ergodic theorems can be formulated, i.e., the convergence of the pointwise averages to the space average. Additionally, a ratio ergodic theorem will be formulated and proven for a large class of dynamical systems.