arXiv subjects

Ronnie Pavlov

Publications and source records attributed to Ronnie Pavlov.

At least 19 recordsLinked to original sources

arXiv2026

Periodic structure and Schrodinger operators for codings of circle rotations

We consider, for any irrational $\alpha$ and interval $I \subset \mathbb{T}$, the 2-interval coding subshift $X^{(I, \alpha)}$ induced by coding orbits under repeated rotation by $\alpha$ via membership in $I$ or $I^c$. Each sequence $c \in X^{(I, \alpha)}$ has an associated Schr\"{o}dinger operator $H_c$, and in \cite{kaminaga} it was proved that if the continued fraction of $\alpha$ has digits with limsup at least $4$, then almost every $c \in X^{(I, \alpha)}$ has so-called $3$-block Gordon structure, which implies that the operator $H_c$ has no eigenvalues. We significantly improve this result by completely characterizing almost-sure $3$-block Gordon structure, proving that in fact it holds for all $(\alpha,I)$ except for a countable set of pairs $(\alpha, |I|)$ where $\alpha$ is M\"{o}bius equivalent to the silver mean and $|I| \in \mathbb{Z}\alpha + f(\alpha)$ where $f(\alpha)$ is a specific infinite series taking value either $\frac{1}{2}, \frac{\alpha}{2}$, or $\frac{\alpha+1}{2}$. We also show that for a set of $\alpha$ of full measure, and for every $I$, the set of points whose orbit codings do not have $3$-block Gordon structure has Hausdorff dimension bounded away from $1$.

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arXiv2026

Stable covers of subshifts

Given a dynamical system, a characteristic measure is a Borel probability measure invariant under all of its automorphisms. Frisch and Tamuz asked if every symbolic system supports such a measure. Motivated by this problem, we study the natural cover of a subshift by its shift of finite type approximations and two senses in which this cover can be said to stabilize. The first is in terms of entropy decay and the second in terms of periodic points. We show that the first type of stabilization gives a new characterization of the class of language stable shifts and demonstrates that there is a mechanism for producing a characteristic measures that relies only on entropy differences. For the second type of stabilization, we show that this defines a new class of subshifts, invariant under conjugacies, that have characteristic measures.

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arXiv2025

On subshifts with low maximal pattern complexity

For a finite alphabet $\mathcal{A}$ and a sequence $x \in \mathcal{A}^{\mathbb{N}}$, Kamae and Zamboni defined the maximal pattern complexity function $p^*_x(n)$ as a natural generalization of usual word complexity. They defined a nonperiodic sequence $x$ to be pattern Sturmian if it achieves the minimal growth rate $p^*_x(n) = 2n$, and asked the question of whether one could classify recurrent pattern Sturmian sequences. We answer their question by characterizing recurrent pattern Sturmian sequences as one of two known types: either a coding of an irrational circle rotation by two intervals, or an element of what we call a nearly simple Toeplitz subshift. We also show that nonrecurrent pattern Sturmian sequences are either very close to constant (such examples were given by Kamae and Zamboni) or a (nonrecurrent) coding of an irrational circle rotation by two intervals. Our main new technique is to use topological properties of the maximal equicontinuous factor (MEF) of the subshift generated by $x$. In this way, we prove a general structural result about sequences with non-superlinear maximal pattern complexity: they are either nonrecurrent or minimal with MEF either an odometer or the product of a circle with a finite cyclic group.

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arXiv2024

Interpolation sets for dynamical systems

Originating in harmonic analysis, interpolation sets were first studied in dynamics by Glasner and Weiss in the 1980s. A set $S \subset \mathbb{N}$ is an interpolation set for a class of topological dynamical systems $\mathcal{C}$ if any bounded sequence on $S$ can be extended to a sequence that arises from a system in $\mathcal{C}$. In this paper, we provide combinatorial characterizations of interpolation sets for: $\bullet$ (totally) minimal systems; $\bullet$ topologically (weak) mixing systems; $\bullet$ strictly ergodic systems; and $\bullet$ zero entropy systems. Additionally, we prove some results on a slightly different notion, called weak interpolation sets, for several classes of systems. We also answer a question of Host, Kra, and Maass concerning the connection between sets of pointwise recurrence for distal systems and $IP$-sets.

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arXiv2024

Computability of Pressure for Subshifts on Countable Amenable Groups

There are a variety of results in the literature proving forms of computability for topological entropy and pressure on subshifts. In this work, we prove two quite general results, showing that topological pressure is always computable from above given an enumeration for a forbidden list inducing the subshift, and that for strongly irreducible shifts of finite type, topological pressure is computable. Our results apply to subshifts on all finitely generated amenable groups with decidable word problem and generalize several previous results which applied only to $\mathbb{Z}^d$-subshifts. As corollaries, we obtain some results related to ground state energy and entropy, proving that the map sending $ϕ$ to $\sup_{μ\in M_σ(X)} \int ϕdμ$ is computable/computable from above when $P_X(ϕ)$ is, and that the map sending $ϕ$ to its ground state/residual entropy is computable from above when $P_X(ϕ)$ is computable. We conclude by giving explicit bounds on computation time of $P_X(ϕ)$ in the $\mathbb{Z}^d$ setting for SI SFTs and locally constant and rational valued $ϕ$, and show that in the special case $X = A^{\mathbb{Z}^2}$, this algorithm runs in singly exponential time.

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arXiv2023

On minimal subshifts of linear word complexity with slope less than 3/2

We prove that every infinite minimal subshift with word complexity $p(q)$ satisfying $\limsup p(q)/q < 3/2$ is measure-theoretically isomorphic to its maximal equicontinuous factor; in particular, it has measurably discrete spectrum. Among other applications, this provides a proof of Sarnak's conjecture for all subshifts with $\limsup p(q)/q < 3/2$ (which can be thought of as a much stronger version of zero entropy). As in \cite{creutzpavlov}, our main technique is proving that all low-complexity minimal subshifts have a specific type of representation via a sequence $\{τ_k\}$ of substitutions, usually called an S-adic decomposition. The maximal equicontinuous factor is the product of an odometer with a rotation on a compact abelian connected one-dimensional group, for which we can give an explicit description in terms of the substitutions $τ_k$. We also prove that all such odometers and groups may appear for minimal subshifts with $\limsup p(q)/q = 1$, demonstrating that lower complexity thresholds do not further restrict the possible structure.

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arXiv2023

Low Complexity Subshifts have Discrete Spectrum

We prove results about subshifts with linear (word) complexity, meaning that $\limsup \frac{p(n)}{n} < \infty$, where for every $n$, $p(n)$ is the number of $n$-letter words appearing in sequences in the subshift. Denoting this limsup by $C$, we show that when $C < \frac{4}{3}$, the subshift has discrete spectrum, i.e. is measurably isomorphic to a rotation of a compact abelian group with Haar measure. We also give an example with $C = \frac{3}{2}$ which has a weak mixing measure. This partially answers an open question of Ferenczi, who asked whether $C = \frac{5}{3}$ was the minimum possible among such subshifts; our results show that the infimum in fact lies in $[\frac{4}{3}, \frac{3}{2}]$. All results are consequences of a general S-adic/substitutive structure proved when $C < \frac{4}{3}$.

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arXiv2023

The extended Hausdorff dimension spectrum of a conformal iterated function system is maximal

For any conformal iterated function system (CIFS) consisting of finitely or countably many maps, and any closed shift-invariant set of right-infinite sequences of such maps, one can associate a limit set, which we call a shift-generated conformal iterated construction. We define the extended Hausdorff dimension spectrum of a CIFS to be the set of Hausdorff dimensions of all such limit sets. We prove that for any CIFS with finitely or countably many maps, the extended Hausdorff dimension spectrum is maximal, i.e. all nonnegative dimensions less than or equal to the dimension of the limit set of the CIFS are realized. We also prove a version of this result even for so-called conformal graph directed Markov systems, obtained via nearest-neighbor restrictions on the CIFS. %when there are nearest-neighbor restrictions on the CIFS (similar to those in the so-called conformal graph directed Markov systems). The main step of the proof is to show that for the family $(X_β)$ of so-called $β$-shifts, the Hausdorff dimension of the limit set associated to $X_β$ varies continuously as a function of $β$.

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arXiv2023

Minimal zero entropy subshifts can be unrestricted along any sparse set

We present a streamlined proof of a result essentially present in previous work of the author, namely that for every set $S = \{s_1, s_2, \ldots\} \subset \mathbb{N}$ of zero Banach density and finite set $A$, there exists a minimal zero-entropy subshift $(X, \sigma)$ so that for every sequence $u \in A^\mathbb{Z}$, there is $x_u \in X$ with $x_u(s_n) = u(n)$ for all $n \in \mathbb{N}$. Informally, minimal deterministic sequences can achieve completely arbitrary behavior upon restriction to a set of zero Banach density. As a corollary, this provides counterexamples to the Polynomial Sarnak Conjecture which are significantly more general than some recently provided in word of Kanigowski, Lema\'{n}czyk, and Radziwi\l\l and of Lian and Shi, and shows that no similar result can hold under only the assumptions of minimality and zero entropy.

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arXiv2023

Measures of maximal entropy of bounded density shifts

We find sufficient conditions for bounded density shifts to have a unique measure of maximal entropy. We also prove that every measure of maximal entropy of a bounded density shift is fully supported. As a consequence of this, we obtain that bounded density shifts are surjunctive.

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arXiv2022

Measure-Theoretically Mixing Subshifts with Low Complexity

We introduce a class of rank-one transformations, which we call extremely elevated staircase transformations. We prove that they are measure-theoretically mixing and, for any $f : \mathbb{N} \to \mathbb{N}$ with $f(n)/n$ increasing and $\sum 1/f(n) < \infty$, that there exists an extremely elevated staircase with word complexity $p(n) = o(f(n))$. This improves the previously lowest known complexity for mixing subshifts, resolving a conjecture of Ferenczi.

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arXiv2022

On the structure of generic subshifts

We investigate generic properties (i.e. properties corresponding to residual sets) in the space of subshifts with the Hausdorff metric. Our results deal with four spaces: the space $\mathbf{S}$ of all subshifts, the space $\mathbf{S}^{\prime}$ of non-isolated subshifts, the closure $\overline{\mathbf{T}^{\prime}}$ of the infinite transitive subshifts, and the closure $\overline{\mathbf{T}\mathbf{T}^{\prime}}$ of the infinite totally transitive subshifts. In the first two settings, we prove that generic subshifts are fairly degenerate; for instance, all points in a generic subshift are biasymptotic to periodic orbits. In contrast, generic subshifts in the latter two spaces possess more interesting dynamical behavior. Notably, generic subshifts in both $\overline{\mathbf{T}^{\prime}}$ and $\overline{\mathbf{T}\mathbf{T}^{\prime}}$ are zero entropy, minimal, uniquely ergodic, and have word complexity which realizes any possible subexponential growth rate along a subsequence. In addition, a generic subshift in $\overline{\mathbf{T}^{\prime}}$ is a regular Toeplitz subshift which is strongly orbit equivalent to the universal odometer.

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arXiv2022

Subsystem entropies of shifts of finite type and sofic shifts on countable amenable groups

In this work we study the entropies of subsystems of shifts of finite type (SFTs) and sofic shifts on countable amenable groups. We prove that for any countable amenable group $G$, if $X$ is a $G$-SFT with positive topological entropy $h(X) > 0$, then the entropies of the SFT subsystems of $X$ are dense in the interval $[0, h(X)]$. In fact, we prove a "relative" version of the same result: if $X$ is a $G$-SFT and $Y \subset X$ is a subshift such that $h(Y) < h(X)$, then the entropies of the SFTs $Z$ for which $Y \subset Z \subset X$ are dense in $[h(Y), h(X)]$. We also establish analogous results for sofic $G$-shifts.

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arXiv2021

Local finiteness and automorphism groups of low complexity subshifts

We prove that for any transitive subshift $X$ with word complexity function $c_n(X)$, if $\liminf \frac{\log (c_n(X)/n)}{\log \log \log n} = 0$, then the quotient group $\textrm{Aut}(X,σ) / \langle σ\rangle$ of the automorphism group of $X$ by the subgroup generated by the shift $σ$ is locally finite. We prove that significantly weaker upper bounds on $c_n(X)$ imply the same conclusion if the Gap Conjecture from geometric group theory is true. Our proofs rely on a general upper bound for the number of automorphisms of $X$ of range $n$ in terms of word complexity, which may be of independent interest. As an application, we are also able to prove that for any subshift $X$, if $\frac{c_n(X)}{n^2 (\log n)^{-1}} \rightarrow 0$, then $\textrm{Aut}(X,σ)$ is amenable, improving a result of Cyr and Kra. In the opposite direction, we show that for any countable infinite locally finite group $G$ and any unbounded increasing $f: \mathbb{N} \rightarrow \mathbb{N}$, there exists a minimal subshift $X$ with $\textrm{Aut}(X,σ) / \langle σ\rangle$ isomorphic to $G$ and $\frac{c_n(X)}{nf(n)} \rightarrow 0$.

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arXiv2020

Ubiquity of entropies of intermediate factors

We consider topological dynamical systems $(X,T)$, where $X$ is a compact metrizable space and $T$ denotes an action of a countable amenable group $G$ on $X$ by homeomorphisms. For two such systems $(X,T)$ and $(Y,S)$ and a factor map $π: X \rightarrow Y$, an intermediate factor is a topological dynamical system $(Z,R)$ for which $π$ can be written as a composition of factor maps $ψ: X \rightarrow Z$ and $φ: Z \rightarrow Y$. In this paper we show that for any countable amenable group $G$, for any $G$-subshifts $(X,T)$ and $(Y,S)$, and for any factor map $ π:X \rightarrow Y$, the set of entropies of intermediate subshift factors is dense in the interval $[h(Y,S), h(X,T)]$. As a corollary, we also prove that if $(X,T)$ and $(Y,S)$ are zero-dimensional $G$-systems, then the set of entropies of intermediate zero-dimensional factors is equal to the interval $[h(Y,S), h(X,T)]$. Our proofs rely on a generalized Marker Lemma that may be of independent interest.

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arXiv2019

On subshifts with slow forbidden word growth

In this work, we treat subshifts, defined in terms of an alphabet $A$ and (usually infinite) forbidden list $F$, where the number of $n$-letter words in $F$ has "slow growth rate" in $n$. We show that such subshifts are well-behaved in several ways; for instance, they are boundedly supermultiplicative as defined by Baker and Ghenciu and they have unique measures of maximal entropy with the K-property and which satisfy Gibbs bounds on large (measure-theoretically) sets. The main tool in our proofs is a more general result which states that bounded supermultiplicativity and a sort of measure-theoretic specification property together imply uniqueness of MME and our Gibbs bounds. We also show that some well-known classes of subshifts can be treated by our results, including the symbolic codings of f(x) = $α+ βx$ (the so-called $α$-$β$ shifts) and the bounded density subshifts of Stanley.

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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.

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arXiv2019

Subsystems of transitive subshifts with linear complexity

We bound the number of distinct minimal subsystems of a given transitive subshift of linear complexity, continuing work of Ormes and Pavlov [7]. We also bound the number of generic measures such a subshift can support based on its complexity function. Our measure-theoretic bounds generalize those of Boshernitzan [1] and are closely related to those of Cyr and Kra [2].

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