Torus Maps and Cocyclic Subshifts

Montana State University, Bozeman MT

Investigators

Abstract

PI: Jarek Kwapisz, Montana State University DMS-0201600 Torus Maps and Cocyclic Subshifts The PI proposes research in two areas: rotation sets for maps of tori and symbolic dynamics of cocyclic subshifts. The first is motivated by classical questions about the flow on invariant tori in Hamiltonian systems or systems of coupled relaxation oscillators and concentrates on maps of a two-torus that are isotopic to the identity. The rotation set is a compact and convex subset of the plane that roughly collects all the average winding vectors (frequencies) exhibited by the orbits of the system. The PI's goal is to establish results on a priori degeneracy of the rotation set to a single point and to understand the extent to which the dynamics must then be equivalent to that of the rigid rotation. The tools involved include the number theory of the rotation set, hierarchies of dynamical tilings, geometric topology, quasi-conformal estimates, and renormalization techniques. Cocyclic subshifts are a new class of algebraically defined dynamical systems obtained as the supports of locally constant matrix cocycles on the full shift over a finite alphabet. They generalize sofic systems and play a role in the symbolic dynamics based on the Conley index similar to the role of subshifts of finite type in the context of Markov partitions. The PI will study the interplay between the algebra of the cocycle and the dynamics of its subshift. Nonlinear oscillatory behavior is typical of many systems encountered in science and technology. The examples include plasma particles (in magnetic fields of the sun or plasma containment devices), electrical currents (in electronics or living tissue), Belousov-Zhabotinsky type chemical reactions, convection currents (in a heated fluid), and planets of our solar system. Many realistic models are impossible to solve analytically and may be accurately followed on a computer only for a limited time. The theory of rotation sets tries to remedy this problem by studying the fundamental properties of systems of coupled oscillators in an abstract setting so that qualitative predictions of the long time behavior can be made for broad classes of models. Symbolic dynamics deals with infinite sequences of zeros and ones and is motivated by problems of telecommunication, where routinely data is encoded by such sequences and the volume is huge enough to justify the idealization to infinite sequences. Loosely speaking, the main goal is to understand the relation between the properties of the sequences and the design of the (mathematical or physical) device generating them. The cocyclic subshifts arise from a new kind of device capable of matrix multiplication. They generalize the so called subshifts of finite type, which are generated by finite state automata and are a standard tool in coding information for magnetic recording. Cocyclic subshifts are also used to detect chaos in dynamical systems.