Thirty Years of Turnstiles and Transport

TL;DR

This survey reviews thirty years of research on transport in deterministic dynamical systems, focusing on measures like fluxes, barriers, and models explaining transport phenomena.

Contribution

It synthesizes progress on computing transport measures, analyzing barriers, and applying Markov models to understand transport dynamics in volume-preserving maps.

Findings

Invariant manifolds form partial barriers impeding transport.

Fluxes can be computed via differences in orbit actions.

Markov models explain algebraic decay in transport distributions.

Abstract

To characterize transport in a deterministic dynamical system is to compute exit time distributions from regions or transition time distributions between regions in phase space. This paper surveys the considerable progress on this problem over the past thirty years. Primary measures of transport for volume-preserving maps include the exiting and incoming fluxes to a region. For area-preserving maps, transport is impeded by curves formed from invariant manifolds that form partial barriers, e.g., stable and unstable manifolds bounding a resonance zone or cantori, the remnants of destroyed invariant tori. When the map is exact volume preserving, a Lagrangian differential form can be used to reduce the computation of fluxes to finding a difference between the action of certain key orbits, such as homoclinic orbits to a saddle or to a cantorus. Given a partition of phase space into regions bounded by partial barriers, a Markov tree model of transport explains key observations, such as the algebraic decay of exit and recurrence distributions.