2008 Bowen Lectures

The Twenty-Eighth Annual Bowen Lectures will be delivered by Hillel Furstenberg on November 18, 19 and 20.

The Bowen lectures are supported by an anonymous donor, who was an undergraduate student of Rufus Bowen.

Department of Mathematics, University of California, Berkeley, presents

The 2008 Bowen Lectures
Hillel Furstenberg
Hebrew University Jerusalem

Ergodic Fractal Measures

Abstract:

These lectures will focus on the role of ergodic theory in the geometry of fractals. We shall be looking at dynamical systems in which progression in time corresponds to progressively increasing magnification of fractals in euclidean space. From this point of view the phenomenon of self-similarity for special fractals can be regarded as corresponding to that of periodicity of orbits in dynamical systems. The more general dynamical phenomena of almost periodicity and recurrence also have their counterparts in the geometry of fractals, and much of our discussion will be devoted to clarifying this. It will be convenient to deal with "fractal measures", i.e., measures supported on fractal sets, for which tools of ergodic theory will be available. These ideas will find application in questions involving Hausdorff dimension, but we will see that "ergodic fractal measures" are objects of independent interest.

Lecture I: Measures exhibiting ergodic properties
Tuesday, November 18
Room F295, Haas - 4:10pm-5:00pm

Abstract: We will begin by describing "ergodic-like" properties of occupation time measures defined on euclidean space of dimension > 2 by the typical transient brownian motion path on the space: "zooming in" to a typical point of such a measure one "covers" the entire range of possibilities. The bulk of our subsequent discussion will provide a mechanism for reproducing this phenomenon. These will be our "ergodic fractal measures". In this first lecture we also describe some of the by-products of this machinery.

Reception in 1015 Evans at 5:15 pm following Tuesday's lecture.

Lecture II: Measure-valued Markov processes and the fractals they generate
Wednesday, November 19
Room 10, Evans Hall - 4:10pm-5:00pm

Abstract: We review the connection between stationary processes and measure preserving dynamical systems. We'll introduce a family of naturally defined measure valued processes ("CP processes"). Ergodic theorems as well as multiple recurrence theorems combined with information-theoretic notions translate to familiar notions on dimension and self-similarity.

Lecture III: Sets, Micro-sets,Galleries, and the CP systems these support
Thursday, November 20
Room F295, Haas - 4:10-5:00pm

Abstract: "Zooming in" on a set in euclidean space gives us the notion of a "mini-set" and limits of these are "micro-sets". The family of micro-sets of a given set forms a "gallery" and a fundamental result is the existence of a CP process whose measures are supported on a gallery. In this final lecture we indicate the proof of this and we examine the implications for infinite and finite subsets in euclidean space.