2012/10/08 by Michael Blank
Mathematics · Physics and Astronomy · #Applied mathematics #Entropy (arrow of time) #Ergodic theory #Ergodicity #Invariant (physics) #Invariant measure #Lattice (music) #Limiting #Markov Chains and Monte Carlo Methods #Markov chain #Markov process #Mathematical Dynamics and Fractals #Mathematics #Pure mathematics #Statistical physics #Statistics #Stochastic process #Stochastic processes and statistical mechanics #Topological entropy #cond-mat.stat-mech #math.DS #math.PR #msc:28D05 #msc:28D20 #msc:34F05 #msc:35B35 #msc:37A50
paper · pdf · doi:10.1088/0951-7715/25/12/3389
23 pages, accepted by "Nonlinearity"
arxiv created 2012/10/08 · openalex publication_date 2012/10/26 · arxiv updated 2015/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
We study the 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 construct (spatially) Markov invariant measures directly. In distinction to conventional results the limiting measures in the non-lattice case are non-ergodic. The average velocity of individual 'vehicles' as a function of their density and its stochastic stability is studied as well.