Abstract
We show how geometric methods from the general theory of fractal dimensions and iterated function systems can be deployed to study symbolic dynamics in the zero entropy regime. More precisely, we establish a dimensional characterization of the topological notion of amorphic complexity. For subshifts with discrete spectrum associated to constant length substitutions, this characterization allows us to derive bounds for the amorphic complexity by interpreting the subshift as the attractor of an iterated function system in a suitable quotient space. As a result, we obtain the general finiteness and positivity of amorphic complexity in this setting and provide a closed formula in case of a binary alphabet.
Highlights
The relation between the dimension theory of dynamical systems and ergodic theory is nowadays a well-established field of research
In this article we provide this kind of relation for the topological notion of amorphic complexity which was recently introduced in [17] to study dynamical systems with zero entropy
An interesting consequence of Theorem 1.1 is that the amorphic complexity of infinite substitution subshifts over two symbols with discrete spectrum is always finite and bounded from below by one
Summary
The relation between the dimension theory of dynamical systems and ergodic theory is nowadays a well-established field of research. Of iterated function systems on general complete metric spaces to estimate the amorphic complexity of subshifts with discrete spectrum associated to primitive constant length substitutions, see Sect. An interesting consequence of Theorem 1.1 is that the amorphic complexity of infinite substitution subshifts over two symbols with discrete spectrum is always finite and bounded from below by one. As it turns out, this holds true over general alphabets. We want to point out that the proof of the previous theorem yields means to compute concrete lower and upper bounds for the amorphic complexity of infinite substitutive subshifts with discrete spectrum, see Sect.
Sign-in/Register to view highlights & summary 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.
