Convergence of Gibbs measures and the behavior of shrinking tubular neighborhoods of fractals and algebraic sets

Annealing is a physical process that motivates our definition of a Gibbs measure, which is a certain probability measure on Euclidean space. In this paper we examine a sequence of Gibbs measures characterized by the distance function. In Chapter 2 we conclude that the sequence of measures converge to a Hausdorff probability measure equally distributed along self-similar fractals with Hutchinson's Open Set Condition. In Chapter 3 we study spaces of concentric circles (which we call targets) in the plane, and examine how the sequence of probability measures distributes over the targets. By varying the number of targets and the size of the circles, we find probability measures that divide their mass between different point masses and spaces. Finally, in Chapter 4 we conclude that the measure will distribute evenly over the highest-dimensional strata of any semi-algebraic set.

Name thesis:degree_name

Doctor of Philosophy

Level thesis:degree_level

Doctoral

Discipline thesis:degree_discipline

Natural Sciences

Grantor

Rice University

Year dc:date.issued

2007

Author dc:creator

• Samansky, Eric Michael

Advisor dc:contributor.advisor

• Hardt, Robert M.

• Mathematics

Statement dc:rights

• Copyright 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.

Language dc:language.iso

eng

Handle dc:identifier.uri

https://hdl.handle.net/1911/20643

OAI identifier oai:identifier

oai:repository.rice.edu:1911/20643