arXiv subjects

E. Arthur Robinson Jr

Publications and source records attributed to E. Arthur Robinson Jr.

2 recordsLinked to original sources

arXiv2025

Directional Expansiveness for Rd-Actions and for Penrose Tilings

We define and study two kinds of directional expansiveness, weak and strong, for an action T of \mathbb{R}^d on a compact metric space X. We show that for \mathbb{R}^2 finite local complexity (FLC) tiling dynamical systems, weak and strong expansiveness are the same, and are both equivalent to a simple coding property. Then we show for the Penrose tiling dynamical system, which is FLC, there are exactly five non expansive directions, the directions perpendicular to the 5th roots of unity. We also study Raphael Robinson's set of 24 Penrose Wang tiles and show the corresponding Penrose Wang tile dynamical system is strictly ergodic. Finally, we study two deformations of the Penrose Wang tile system, one where the square Wang tiles are all deformed into a 2π/5 rhombus, and another where they are deformed into a set of eleven tetragon tiles. We show both of these are topologically conjugate to the Penrose tiling dynamical system.

math.DS↗

arXiv2014

Parry's topological transitivity and f-expansions

In his 1964 paper on f-expansions, Parry studied piecewise-continuous, piecewise-monotonic maps F of the interval [0,1), and introduced a notion of topological transitivity different from any of the modern definitions. This notion, which we call Parry topological transitivity, (PTT) is that the backward orbit O^-(x)={y:x=F^ny for some n\ge 0} of some x\in[0,1) is dense. We take topological transitivity (TT) to mean that some $x$ has a dense forward orbit. Parry's application to f-expansions is that PTT implies the partition of [0,1) into the "fibers" of F is a generating partition (i.e., f-expansions are "valid"). We prove the same result for TT, and use this to show that for interval maps F, TT implies PTT. A separate proof is provided for continuous maps F of compact metric spaces. The converse is false.

math.DS↗