Automata, Dynamics and Actions

Introduction

This project lies at the nexus of complex and symbolic dynamics, group theory, decision problems, and computation. It aims to solve major problems in each of these fields by means of automatic actions and relations.

Finite State Automata

Finite state automata, pervasive in theoretical computer science, will serve to define self-similar mathematical objects and produce efficient algorithms to manipulate them. I will explore a novel notion of automatically acting group, encompassing the previously unrelated notions of automatic groups, automata groups, and substitutive shifts.

Geometric Group Theory

Geometric group theory propounds the vision of groups as geometric objects. A basic notion is volume growth, and Milnor's still open gap problem asks for its possible range. In this proposal, I will give candidates of groups with very slow superpolynomial growth, defined by their automatic action on dynamical systems, and a proof strategy.

Sofic Groups

A celebrated open problem by Gromov asks whether all groups are sofic. This property has too many valuable consequences to always be true, yet there is no known non-example! I will present a strategy of producing non-sofic groups closely associated with automata.

Rational Maps on the Riemann Sphere

Rational maps on the Riemann sphere provide a rich supply of dynamical systems. A fundamental goal is to give a combinatorial description of the dynamics across families of maps, constructing models of parameter space. I will encode the maps via automatic actions and study relations between automata to produce such models. I aim to achieve a full topological description (including the long-open connectedness problem) of Milnor's slices of quadratic maps.

Conclusion

This project will tackle these fundamental questions from group theory and dynamics and develop presently unexplored interactions between them through a unified use of automata. It will prove decidability of certain algorithmic problems such as Dehn's and Tarski's, and construct efficient tools to further our exploration of these mathematical universes.