Complex Dynamics and Symbolic Dynamics

Robert L. Devaney, Boston University Synopsis As so often happens in mathematics, there is a surprising connection between two quite distinct subfields of dynamical systems theory, namely the structure of the automorphism group of the one sided shift map on d-symbols and the topology of the analogue of the Mandelbrot set for degree d polynomials of one complex variable. In this lecture we will give an elementary overview of both of these topics, highlighting the tools that relate them. Let Σd denote the space of (one-sided) sequences of integers 0, 1, . . . d − 1 and let σ denote the usual shift map σ(s0s1s2 . . . ) = (s1s2 . . . ). An important question in symbolic dynamics concerns the group of automorphisms of the shift, i.e., maps η : Σd → Σd that commute with the shift map. For one-sided shift maps, this group is well understood thanks to work of Hedlund [6], Boyle, Franks, and Kitchens [3], and Ashley [1]. The automorphism group for the 2-shift is simple: There is only one non-trivial element, namely the automorphism that interchanges the two symbols 0 and 1. For the d-shift, the group is infinitely generated with a rich algebraic structure. Turning now to complex dynamics, consider first the dynamics of quadratic polynomials of the form Qc(z) = z + c. As is well known, the interesting dynamics of this map takes place on the Julia set [7], Jc. This set assumes one of two topological types: Either Jc is connected or Jc is a Cantor set. In the latter case, the action of Qc on the Julia set is equivalent to the one-sided 2-shift. The well known Mandelbrot set M is a picture of this dichotomy: If c lies in M , the Julia set is connected; outside M , Jc is a Cantor set. If we follow a closed loop in the complement of M , the return to the original position induces an automorphism of the shift. If this loop winds once around M , then the induced autormorphism is the non-trivial element of the group; on the other hand, if the loop does not contain M , the trivial automorphism is induced. The main goal of this lecture is to describe a similar phenomenon that occurs for polynomials of higher degree. Here the analogue of M lies in complex d− 1 dimensional space and has a rich topology. Following loops around various portions of this space again induces an automorphism of the d-shift. We will describe how one can generate every automorphism of the shift in this manner, thus yielding a surjection from the fundamental group of this space onto the group of automorphisms [2]. For background on complex dynamics, we suggest the Proceedings from several previous AMS Short Courses [4],[5]. For background on the relevant symbolic dynamics, see [8].