arXiv subjects

Daniel Sell

Publications and source records attributed to Daniel Sell.

6 recordsLinked to original sources

arXiv2024

Almost automorphic subshifts with finiteness conditions for the boundary of the separating cover

In this article we study orbits of proximal pairs in almost automorphic subshifts. The corresponding orbits in the maximal equicontinuous factor are precisely those orbits that intersect the boundary of the subshift's separating cover. We impose certain finiteness conditions on this boundary and explore the resulting consequences for the subshift, for instance in terms of complexity or the relations between proximal and asymptotic pairs. The last part of our article deals with Toeplitz subshifts without a finite boundary. There we treat the question of necessary conditions and sufficient conditions for the existence of a factor subshift with a finite boundary. Throughout the whole article, we provide explicit constructions for Toeplitz subshifts to illustrate our findings and the necessity of our assumptions.

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arXiv2023

Invariant measures for $\mathscr{B}$-free systems revisited

For $ \mathscr{B} \subseteq \mathbb{N} $, the $ \mathscr{B} $-free subshift $ X_{\eta} $ is the orbit closure of the characteristic function of the set of $ \mathscr{B} $-free integers. We show that many results about invariant measures and entropy, previously only known for the hereditary closure of $ X_{\eta} $, have their analogues for $ X_{\eta} $ as well. In particular, we settle in the affirmative a conjecture of Keller about a description of such measures ([Keller, G. Generalized heredity in $\mathcal B$-free systems. Stoch. Dyn. 21, 3 (2021), Paper No. 2140008]). A central assumption in our work is that $ \eta^{*} $ (the Toeplitz sequence that generates the unique minimal component of $ X_{\eta} $) is regular. From this we obtain natural periodic approximations that we frequently use in our proofs to bound the elements in $ X_{\eta} $ from above and below.

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arXiv2021

On the Garden of Eden theorem for B-free subshifts

We prove that on B-free subshifts, with B satisfying the Erd\"os condition, all cellular automata are determined by monotone sliding block codes. In particular, this implies the validity of the Garden of Eden theorem for such systems.

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arXiv2020

Simple Toeplitz subshifts: combinatorial properties and uniformity of cocycles

We investigate combinatorial properties of aperiodic simple Toeplitz subshifts, as well as spectral properties of Jacobi operators defined by them. More precisely, we derive explicit formulas for complexity, palindrome complexity and, for sufficiently large word length, repetitivity. In addition we give a complete description of the de Bruijn graphs. We characterise alpha-repetitivity and, based on a work by Liu and Qu from 2011, the Boshernitzan condition. These combinatorial results can also be found in [arXiv:1801.08778]. Regarding the Jacobi operators, we show that they have empty pure point spectrum for almost all elements in the subshift. This generalises a result of Grigorchuk, Lenz and Nagnibeda from 2018. In addition the spectrum is shown to be a Cantor set of Lebesgue measure zero. In fact we prove the stronger statement that every locally constant SL(2,R)-cocycle is uniform. To do so, we use the so-called leading sequence condition for subshifts, which stems from a collaboration with Grigorchuk, Lenz and Nagnibeda, see [arXiv:1906.01898]. This approach allows us to establish uniformity of cocycles for simple Toeplitz subshifts and Sturmian subshifts in a unified way. An appendix briefly reviews the connection between Jacobi operators on simple Toeplitz subshifts and Laplacians on Schreier graphs of self-similar groups.

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arXiv2019

Subshifts with leading sequences, uniformity of cocycles and spectra of Schreier graphs

We introduce a class of subshifts governed by finitely many two-sided infinite words. We call these words leading sequences. We show that any locally constant cocycle over such a subshift is uniform. From this we obtain Cantor spectrum of Lebesgue measure zero for associated Jacobi operators if the subshift is aperiodic. Our class covers all simple Toeplitz subshifts as well as all Sturmian subshifts. We apply our results to the spectral theory of Schreier graphs for uncountable families of groups acting on rooted trees.

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arXiv2018

Combinatorics of One-Dimensional Simple Toeplitz Subshifts

This paper provides a systematic study of fundamental combinatorial properties of one-dimensional, two-sided infinite simple Toeplitz subshifts. Explicit formulas for the complexity function, the palindrome complexity function and the repetitivity function are proven. Moreover, a complete description of the de Bruijn graphs of the subshifts is given. Finally, the Boshernitzan condition is characterised in terms of combinatorial quantities, based on a recent result of Liu and Qu. Particular simple characterisations are provided for simple Toeplitz subshifts that correspond to the orbital Schreier graphs of the family of Grigorchuk's groups, a class of subshifts that serves as main example throughout the paper.

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