Abstract
A symbolic dynamical system is a continuous transformation Φ : X ⟶ X of closed subset X ⊆ A V , where A is a finite set and V is countable (examples include subshifts, odometers, cellular automata, and automaton networks). The function Φ induces a directed graph (‘network’) structure on V , whose geometry reveals information about the dynamical system ( X , Φ ) . The dimension dim ( V ) is an exponent describing the growth rate of balls in this network as a function of their radius. We show that, if X has positive entropy and dim ( V ) > 1 , and the system ( A V , X , Φ ) satisfies minimal symmetry and mixing conditions, then ( X , Φ ) cannot be positively expansive; this generalizes a well-known result of Shereshevsky about multidimensional cellular automata. We also construct a counterexample to a version of this result without the symmetry condition. Finally, we show that network dimension is invariant under topological conjugacies which are Hölder-continuous.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
