On the convergence of ergodic averages over zero density sequences in topological dynamics.

摘要

The purpose of this project was to study topological analogs of certain theorems in measurable dynamics. In particular, Bourgain's theorem was studied and found not to exist in topological dynamics. This negative outcome is due to a positive result about interpolation in sequence spaces.; If T is an ergodic endomorphism of the probability space {dollar}(Omega, Sigma, mu),{dollar} then according to Bourgain's Theorem, for any {dollar}rm fin Lsp2(Omega, Sigma, mu{dollar}) {dollar}{dollar}rmlimsb{lcub}ntoinfty{rcub} {lcub}1over n{rcub}sumsbsp{lcub}k=0{rcub}{lcub}n-1{rcub}fleft(Tsp{lcub}ksp2{rcub}xright) = intsbOmega f(x)dmu mu{lcub}-{rcub}a.e.{dollar}{dollar}In general, a sequence {dollar}rm {lcub}nsb{lcub}k{rcub}{rcub}{dollar} can replace the sequence {dollar}rm {lcub}ksp2{rcub}{dollar} in these averages. J. Bourgain describes convergence of these averages for many sequences of zero density. Many theorems in measurable dynamics have analogs in topological dynamics. In this paper, a topological analog to Bourgain's result was studied with negative results. We established that if {dollar}tau{dollar} is a uniquely ergodic continuous transformation of a compact metric space (X, d), then we cannot expect that for every {dollar}rm fin C(X){dollar} {dollar}{dollar}rm limsb{lcub}ntoinfty{rcub}{lcub}1over n{rcub}sumsbsp{lcub}k=0{rcub}{lcub}n-1{rcub}fleft(tausp{lcub}ksp2{rcub}xright){dollar}{dollar}exists everywhere in X. Further, we showed that, for any uniquely ergodic transformation {dollar}tau{dollar} on (X, d), the averages {dollar}{dollar}rm {lcub}1over n{rcub}sumsbsp{lcub}k=0{rcub}{lcub}n-1{rcub} fleft(tausp{lcub}nsb{lcub}k{rcub}{rcub}xright){dollar}{dollar}may not converge as long as {dollar}rm {lcub}nsb{lcub}k{rcub}{rcub}{dollar} is a zero density sequence of integers satisfying the property that {dollar}rm nsb{lcub}k+1{rcub} - nsb{lcub}k{rcub}{dollar} is increasing and unbounded.; We negated Bourgain's result in topological dynamics by means of a positive result, dealing with interpolation in symbolic dynamics. The dynamical system will be (X, {dollar}tau),{dollar} where X = {dollar}{lcub}0, 1{rcub}sp{lcub}rm Z{rcub}{dollar} and {dollar}tau{dollar} is the shift transformation. The interpolation result, which is the highlight of this paper, may be stated as follows: If {dollar}rm y = {lcub}y(n){rcub}in X{dollar} and {dollar}rm {lcub}esb{lcub}k{rcub}{rcub}{dollar} is any sequence of integers satisfying the properties that {dollar}rm {lcub}esb{lcub}k{rcub}{rcub}{dollar} is increasing, and {dollar}rm {lcub}esb{lcub}k+1{rcub} - esb{lcub}k{rcub}{rcub}{dollar} is an increasing unbounded sequence, then there exists {dollar}rm x = {lcub}x(n){rcub}in X{dollar} satisfying (1) x is uniquely ergodic, and (2) for each {dollar}rm nin{lcub}bf Z{rcub}, x(esb{lcub}n{rcub}) = y(n).{dollar}