# Mathematics Colloquia and Seminars

The past 30 years saw great activity in the study of dynamical systems. One of the successes of this period was to produce a (nearly) complete description of the dynamical behavior of the {\em quadratic family} $x\mapsto a-x^2$, a family given by a very simple formula yet presenting extremely complicated dynamics. Although a fair amount of work has been done in the study of families of 2-dimensional maps --- a famous example being the {\em H\'enon family} $(x,y)\mapsto (a-by-x^2, x)$, which is again given by a simple formula but presents extremely complicated dynamics --- our understanding of them is not nearly as thorough. In this talk I will present several techniques developed to describe the dynamics of families of 2-dimensional diffeomorphisms. The overall approach is to try to create for 2-dimensional maps analogs of the ideas that were central in explaining 1-dimensional families. I will explain {\em pruning}, which is a 2-dimensional analog of the {\em kneading theory} of Milnor and Thurston. This enables us to construct a family of models for 2-dimensional maps. This family is intimately related to a {\em generalized kneading family} of maps of trees and graphs. It also contains the family of all {\em pseudo-Anosov} homeomorphisms that's reasonable to expect it to. I will also explain the {\em 0-entropy equivalence relation}. It enables us to compare the models of the pruning family to actual maps we are trying to model (those in the H\'enon family, say) and give a precise statement of the {\em Pruning Front Conjecture} of Cvitanovi\'c, Gunaratne and Procaccia. It also produces new --- and rather interesting --- families of 2-dimensional maps: that of {\em generalized pseudo-Anosovs} and that of {\em very generalized pseudo-Anosovs}. These families contain, conjecturally, piecewise linear models for 2-dimensional maps. I will also indicate connections between them, infinite-dimensional Teichm\"uller theory and a completion of certain sets of braids in a Gromov-Haudorff sense.