# 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

**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.**

*Abstract:***Lecture III: Sets, Micro-sets,Galleries, and the CP systems these support**

**Thursday, November 20**

Room F295, Haas - 4:10-5:00pm

**"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.**

*Abstract:*