2014/11/19 by Richard Miles, Thomas Ward · 1 citation
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Algebraic number #Connection (principal bundle) #Dynamical systems theory #Entropy (arrow of time) #Geometry #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Physics #Pure mathematics #Quantum chaos and dynamical systems #Quantum mechanics #Statistical physics #Topological entropy #math.DS #msc:37B40 #msc:37C25 #msc:37C40 #msc:37P35
paper · pdf · doi:10.3934/dcdsb.2015.20.3525
published as Discrete Contin. Dyn. Syst. Ser. B 20 (2015), no. 10, 3525-3545
arxiv created 2014/11/19 · openalex publication_date 2015/01/01 · arxiv updated 2016/10/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Dynamical systems generated by d≥2commuting homeomorphisms (topological ℤd-actions)contain within themstructures on many scales,and in particular containmany actions of ℤk for 1≤ k≤ d.Familiar dynamical invariants forhomeomorphisms, like entropy andperiodic point data, become more complexand permit multiple definitions. We brieflysurvey some of these and other relatedinvariants in the setting ofalgebraic ℤd-actions,showing how, even insettings wherethe natural entropy as a ℤd-actionvanishes, a powerful theory ofdirectional entropyand periodic pointscan be built. An underlying themeis uniformity in dynamicalinvariants as the direction changes,and the connection between thistheory and problems in number theory;we explore this for severalinvariants.We also highlight Fried'snotion of average entropy and itsconnection to uniformities in growthproperties, and prove a new relationshipbetween this entropy and periodic pointgrowth in this setting.