Boshernitzan, Michael2009-06-032009-06-032008Ralston, David. "Heaviness: An extension of a lemma of Yuval Peres." (2008) Diss., Rice University. <a href="https://hdl.handle.net/1911/22199">https://hdl.handle.net/1911/22199</a>.https://hdl.handle.net/1911/22199In a 1988 paper, Y. Peres proved that in any probability-measure preserving system {X, mu, T}, where X is compact and T is continuous, coupled with a continuous function f, there is at least one point x (which we term heavy) such that 1ni=0 n-1f&j0;Ti&parl0;x&parr0;&ge; Xfdm for all n &isin; N . We simplify and expand the proof to a more general result, and investigate the properties of the set of heavy points for a given T and f. The structure of the set of such points in irrational circle rotations is studied extensively, followed by a development of similar ideas in symbolic dynamics. Finally, the notion of heaviness is exported to arbitrary sequences, and a few results contrasting heaviness with equidistribution are developed. Open problems for future research are included throughout.118 p.application/pdfengCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.MathematicsThesisTHESIS MATH. 2008 RALSTON