arXiv subjects
Publications and source records attributed to Lori Alvin.
We investigate minimal bounded speedups of Toeplitz flows. We demonstrate that the minimal bounded speedup of a Toeplitz flow need not be another Toeplitz flow and describe techniques for determining whether the resulting speedup is Toeplitz. We also provide sufficient conditions to guarantee that the minimal bounded speedup will be a Toeplitz flow.
In this paper, we construct families of unimodal maps with wild Cantor attractors and study, via symbolic dynamics, the interplay between recurrence properties of the critical orbit. Using kneading and co-kneading techniques, we provide a symbolic characterization of persistent recurrence in terms of cutting and co-cutting times, and relate it to other recurrence conditions, including the Collet--Eckmann condition. As an application, we obtain uncountable families of non-renormalizable unimodal maps of sufficiently high critical order admitting wild attractors. We also analyze the structure of postcritical dynamics in recurrent regimes, constructing examples with embedded odometers and with minimal homeomorphic dynamics on the critical $ω$-limit set, and showing that a minimal homeomorphism together with regular recurrence does not imply conjugacy to an adding machine. Finally, we prove that longbranched combinatorics is incompatible with persistent recurrence, thereby clarifying the boundary between different recurrence regimes.
We study the properties of folding points and endpoints of unimodal inverse limit spaces. We distinguish between non-end folding points and three types of end-points (flat, spiral and nasty) and give conditions for their existence and prevalence. Additionally, we give a characterisation of tent inverse limit spaces for which the set of folding points equals the set of endpoints.
This paper explores the range of bounded speedups in the topological category. Bounded speedups represent both a strengthening of topological speedups as defined in [A 16] and a generalization of powers of a transformation. Here we show that bounded speedups preserve the structure of two classical minimal Cantor systems. Specifically, a minimal bounded speedup of an odometer is a conjugate odometer, and a minimal bounded speedup of a primitive substitution is again a primitive substitution, though it is never conjugate to the original substitution system. Further, we give bounds on the topological entropy of bounded speedups, and in special cases we compute the topological entropy of bounded speedups.