Stochastic stability of traffic maps

TL;DR

This paper investigates the ergodic properties and stochastic stability of traffic maps modeling vehicle flow, providing new insights into invariant measures, entropy, and vehicle velocity behavior in both deterministic and random settings.

Contribution

It introduces novel methods for constructing invariant measures in traffic maps, including non-ergodic limiting measures in non-lattice cases, and analyzes their stochastic stability.

Findings

Existence of nontrivial invariant measures for traffic maps.

Proven stochastic stability of these measures.

Calculated topological entropy for the system.

Abstract

We study ergodic properties of a family of traffic maps acting in the space of bi-infinite sequences of real numbers. The corresponding dynamics mimics the motion of vehicles in a simple traffic flow, which explains the name. Using connections to topological Markov chains we obtain nontrivial invariant measures, prove their stochastic stability, and calculate the topological entropy. Technically these results in the deterministic setting are related to the construction of measures of maximal entropy via measures uniformly distributed on periodic points of a given period, while in the random setting we directly construct (spatially) Markov invariant measures. In distinction to conventional results the limiting measures in non-lattice case are non-ergodic. Average velocity of individual ``vehicles'' as a function of their density and its stochastic stability is studied as well.