arXiv subjects

Nic Ormes

Publications and source records attributed to Nic Ormes.

3 recordsLinked to original sources

arXiv2019

On the complexity function for sequences which are not uniformly recurrent

We prove that every non-minimal transitive subshift $X$ satisfying a mild aperiodicity condition satisfies $\limsup c_n(X) - 1.5n = \infty$, and give a class of examples which shows that the threshold of $1.5n$ cannot be increased. As a corollary, we show that any transitive $X$ satisfying $\limsup c_n(X) - n = \infty$ and $\limsup c_n(X) - 1.5n < \infty$ must be minimal. We also prove some restrictions on the structure of transitive non-minimal $X$ satisfying $\liminf c_n(X) - 2n = -\infty$, which imply unique ergodicity (for a periodic measure) as a corollary, which extends a result of Boshernitzan from the minimal case to the more general transitive case.

math.DS↗

arXiv2015

Subshifts with Slowly Growing Numbers of Follower Sets

For any subshift, define $F_X(n)$ to be the collection of distinct follower sets of words of length $n$ in $X$. Based on a similar result of the second and third authors, we conjecture that if there exists an $n$ for which $|F_X(n)| \leq n$, then $X$ is sofic. In this paper, we prove several results related to this conjecture, including verifying it for $n \leq 3$, proving that the conjecture is true for a large class of coded subshifts, and showing that if there exists $n$ for which $|F_X(n)| \leq \log_2(n+1)$, then $X$ is sofic.

math.DS↗

arXiv2000

A Homeomorphism Invariant for Substitution Tiling Spaces

We derive a homeomorphism invariant for those tiling spaces which are made by rather general substitution rules on polygonal tiles, including those tilings, like the pinwheel, which contain tiles in infinitely many orientations. The invariant is a quotient of Cech cohomology, is easily computed directly from the substitution rule, and distinguishes many examples, including most pinwheel-like tiling spaces. We also introduce a module structure on cohomology which is very convenient as well as of intuitive value.

math.DS↗