arXiv subjects

Ronnie Pavlov

Publications and source records attributed to Ronnie Pavlov.

At least 37 records · Page 2Linked to original sources

arXiv2019

Extender sets and measures of maximal entropy for subshifts

We prove inequalities relating the measures of maximal entropy of two patterns u,v where the extender set of u is contained in the extender set of v. Our main results are two generalizations of a Theorem of Meyerovitch; the first applies to all such v,w when G=Z, and the second to v,w with the same shape and any countable amenable finitely generated torsion-free G. As a consequence of our results we give new and simpler proofs of several facts about synchronizing subshifts and we answer a question of Climenhaga.

math.DS↗

arXiv2019

The relationship between word complexity and computational complexity in subshifts

We prove several results about the relationship between the word complexity function of a subshift and the set of Turing degrees of points of the subshift, which we call the Turing spectrum. Among other results, we show that a Turing spectrum can be realized via a subshift of linear complexity if and only if it consists of the union of a finite set and a finite number of cones, that a Turing spectrum can be realized via a subshift of exponential complexity (i.e. positive entropy) if and only if it contains a cone, and that every Turing spectrum which either contains degree 0 or is a union of cones is realizable by subshifts with a wide range of 'intermediate' complexity growth rates between linear and exponential.

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arXiv2018

Factoring onto $\mathbb{Z}^d$ subshifts with the finite extension property

We define the finite extension property for $d$-dimensional subshifts, which generalizes the topological strong spatial mixing condition defined by Briceño (2016), and we prove that this property is invariant under topological conjugacy. Moreover, we prove that for every $d$, every $d$-dimensional block gluing subshift factors onto every $d$-dimensional subshift which has strictly lower entropy, a fixed point, and the finite extension property. This result extends a theorem from Boyle, Pavlov, and Schraudner (2010), which requires that the factor contain a safe symbol.

math.DS↗

arXiv2018

On entropy and intrinsic ergodicity of coded subshifts

Any coded subshift X defined by a set C of code words contains a subshift, which we call L, consisting of limits of single code words. We show that when C satisfies a unique decomposition property, the topological entropy h(X) of X is determined completely by h(L) and the number of code words of each length. More specifically, we show that h(X) = h(L) exactly when a certain infinite series is less than or equal to 1, and when that series is greater than 1, we give a formula for h(X_C). In the latter case, an immediate corollary (using results of Climenhaga and Thompson) is that X has a unique measure of maximal entropy.

math.DS↗

arXiv2017

Follower, Predecessor, and Extender Entropies

Using follower/predecessor/extender set sequences, we define quantities which we call the follower/predecessor/extender entropies, which can be associated to any shift space. We analyze the behavior of these quantities under conjugacies and factor maps, most notably showing that extender entropy is a conjugacy invariant and that having follower entropy zero is a conjugacy invariant. We give some applications, including examples of shift spaces with equal entropy which can be distinguished by extender entropy, and examples of shift spaces which can be shown to not be isomorphic to their inverse by using follower/predecessor entropy.

math.DS↗

arXiv2017

One-sided almost specification and intrinsic ergodicity

Shift spaces with the specification property are intrinsically ergodic, i.e. they have a unique measure of maximal entropy. This can fail for shifts with the weaker almost specification property. We define a new property called one-sided almost specification, which lies in between specification and almost specification, and prove that it guarantees intrinsic ergodicity if the corresponding mistake function g is bounded. We also show that uniqueness may fail for unbounded g such as log log n. Our results have consequences for almost specification: we prove that almost specification with g=1 implies one-sided almost specification (with g=1), and hence uniqueness. On the other hand, the second author showed recently that almost specification with g=4 does not imply uniqueness. This leaves open the question of whether almost specification implies intrinsic ergodicity when g=2 or g=3.

math.DS↗

arXiv2017

On non-uniform specification and uniqueness of the equilibrium state in expansive systems

In [2], Bowen showed that for an expansive system (X, T) with specification and a potential with the Bowen property, the equilibrium state is unique and fully supported. We generalize that result by showing that the same conclusion holds for non-uniform versions of Bowen's hypotheses in which constant parameters are replaced by any increasing unbounded functions f(n) and g(n) with sublogarithmic growth (in n). We prove results for two weakenings of specification; the first is non-uniform specification, based on a definition of Marcus in ([14]), and the second is a significantly weaker property which we call non-uniform transitivity. We prove uniqueness of the equilibrium state in the former case under the assumption that liminf (f(n) + g(n))/ln n = 0, and in the latter case when lim (f(n) + g(n))/ln n = 0. In the former case, we also prove that the unique equilibrium state has the K-property. It is known that when f(n)/ln n or g(n)/ln n is bounded from below, equilibrium states may not be unique, and so this work shows that logarithmic growth is in fact the optimal transition point below which uniqueness is guaranteed. Finally, we present some examples for which our results yield the first known proof of uniqueness of equilibrium state.

math.DS↗

arXiv2017

Factor maps and embeddings for random $\mathbb{Z}^d$ shifts of finite type

For any $d \geq 1$, random $\mathbb{Z}^d$ shifts of finite type (SFTs) were defined in previous work of the authors. For a parameter $α\in [0,1]$, an alphabet $\mathcal{A}$, and a scale $n \in \mathbb{N}$, one obtains a distribution of random $\mathbb{Z}^d$ SFTs by randomly and independently forbidding each pattern of shape $\{1,\dots,n\}^d$ with probability $1-α$ from the full shift on $\mathcal{A}$. We prove two main results concerning random $\mathbb{Z}^d$ SFTs. First, we establish sufficient conditions on $α$, $\mathcal{A}$, and a $\mathbb{Z}^d$ subshift $Y$ so that a random $\mathbb{Z}^d$ SFT factors onto $Y$ with probability tending to one as $n$ tends to infinity. Second, we provide sufficient conditions on $α$, $\mathcal{A}$ and a $\mathbb{Z}^d$ subshift $X$ so that $X$ embeds into a random $\mathbb{Z}^d$ SFT with probability tending to one as $n$ tends to infinity.

math.DS↗

arXiv2015

Strong spatial mixing in homomorphism spaces

Given a countable graph $\mathcal{G}$ and a finite graph $\mathrm{H}$, we consider $\mathrm{Hom}(\mathcal{G},\mathrm{H})$ the set of graph homomorphisms from $\mathcal{G}$ to $\mathrm{H}$ and we study Gibbs measures supported on $\mathrm{Hom}(\mathcal{G},\mathrm{H})$ . We develop some sufficient and other necessary conditions on $\mathrm{Hom}(\mathcal{G},\mathrm{H})$ for the existence of Gibbs specifications satisfying strong spatial mixing (with exponential decay rate). We relate this with previous work of Brightwell and Winkler, who showed that a graph $\mathrm{H}$ has a combinatorial property called dismantlability if and only if for every $\mathcal{G}$ of bounded degree, there exists a Gibbs specification with unique Gibbs measure. We strengthen their result by showing that this unique Gibbs measure can be chosen to have weak spatial mixing, but we also show that there exist dismantlable graphs for which no Gibbs measure has strong spatial mixing.

math.CO↗

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↗

arXiv2015

Representation and poly-time approximation for pressure of $\mathbb{Z}^2$ lattice models in the non-uniqueness region

We develop a new pressure representation theorem for nearest-neighbour Gibbs interactions and apply this to obtain the existence of efficient algorithms for approximating the pressure in the $2$-dimensional ferromagnetic Potts, multi-type Widom-Rowlinson and hard-core models. For Potts, our results apply to every inverse temperature but the critical. For Widom-Rowlinson and hard-core, they apply to certain subsets of both the subcritical and supercritical regions. The main novelty of our work is in the latter.

math.DS↗

arXiv2015

Topologically completely positive entropy and zero-dimensional topologically completely positive entropy

In a previous paper ("A characterization of topologically completely positive entropy for shifts of finite type"), the author gave a characterization for when a $\mathbb{Z}^d$-shift of finite type (SFT) has no nontrivial subshift factors with zero entropy, a property which we here call zero-dimensional topologically completely positive entropy (ZTCPE). In this work, we study the difference between this notion and the more classical topologically completely positive entropy (TCPE) of Blanchard. We show that there are one-dimensional subshifts and two-dimensional SFTs which have ZTCPE but not TCPE. In addition, we show that strengthening the hypotheses of the main result of the aforementioned paper yields a sufficient condition for a $\mathbb{Z}^d$-SFT to have TCPE.

math.DS↗

arXiv2014

On intrinsic ergodicity and weakenings of the specification property

Since seminal work of Bowen, it has been known that the specification property implies various useful properties about a topological dynamical system, among them uniqueness of the measure of maximal entropy (often referred to as intrinsic ergodicity). Weakenings of the specification property called almost weak specification and almost specification have been defined and profitably applied in various works. However, it has been an open question whether either or both of these properties imply intrinsic ergodicity. We answer this question negatively by exhibiting examples of subshifts with multiple measures of maximal entropy with disjoint support which have almost weak specification with any gap function $f(n) = O(\ln n)$ or almost specification with any mistake function $g(n) \geq 4$. We also show some results in the opposite direction, showing that subshifts with almost weak specification with gap function $f(n) = o(\ln n)$ or almost specification with mistake function $g(n) = 1$ cannot have multiple measures of maximal entropy with disjoint support.

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arXiv2014

Random $\mathbb{Z}^d$-shifts of finite type

In this work we consider an ensemble of random $\mathbb{Z}^d$-shifts of finite type ($\mathbb{Z}^d$-SFTs) and prove several results concerning the behavior of typical systems with respect to emptiness, entropy, and periodic points. These results generalize statements made in \cite{McGoff} regarding the case $d=1$. Let $\mathcal{A}$ be a finite set, and let $d \geq 1$. For $n$ in $\mathbb{N}$ and $α$ in $[0,1]$, define a random subset $ω$ of $\mathcal{A}^{[1,n]^d}$ by independently including each pattern in $\mathcal{A}^{[1,n]^d}$ with probability $α$. Let $X_ω$ be the (random) $\mathbb{Z}^d$-SFT built from the set $ω$. For each $α\in [0,1]$ and $n$ tending to infinity, we compute the limit of the probability that $X_ω$ is empty, as well as the limiting distribution of entropy of $X_ω$. Furthermore, we show that the probability of obtaining a nonempty system without periodic points tends to zero. For $d>1$, the class of $\mathbb{Z}^d$-SFTs is known to contain strikingly different behavior than is possible within the class of $\mathbb{Z}$-SFTs. Nonetheless, the results of this work suggest a new heuristic: typical $\mathbb{Z}^d$-SFTs have similar properties to their $\mathbb{Z}$-SFT counterparts.

math.DS↗

arXiv2014

On intrinsic ergodicity of factors of $\mathbb{Z}^d$ subshifts

It is well-known that any $\mathbb{Z}$ subshift with the specification property has the property that every factor is intrinsically ergodic, i.e., every factor has a unique factor of maximal entropy. In recent work, other $\mathbb{Z}$ subshifts have been shown to possess this property as well, including $β$-shifts and a class of $S$-gap shifts. We give two results that show that the situation for $\mathbb{Z}^d$ subshifts with $d >1 $ is quite different. First, for any $d>1$, we show that any $\mathbb{Z}^d$ subshift possessing a certain mixing property must have a factor with positive entropy which is not intrinsically ergodic. In particular, this shows that for $d>1$, $\mathbb{Z}^d$ subshifts with specification cannot have all factors intrinsically ergodic. We also give an example of a $\mathbb{Z}^2$ shift of finite type, introduced by Hochman, which is not even topologically mixing, but for which every positive entropy factor is intrinsically ergodic.

math.DS↗

arXiv2014

An integral representation for topological pressure in terms of conditional probabilities

Given an equilibrium state $μ$ for a continuous function $f$ on a shift of finite type $X$, the pressure of $f$ is the integral, with respect to $μ$, of the sum of $f$ and the information function of $μ$. We show that under certain assumptions on $f$, $X$ and an invariant measure $ν$, the pressure of $f$ can also be represented as the integral with respect to $ν$ of the same integrand. Under stronger hypotheses we show that this representation holds for all invariant measures $ν$. We establish an algorithmic implication for approximation of pressure, and we relate our results to a result in thermodynamic formalism.

math.DS↗

arXiv2013

On independence and entropy for high-dimensional isotropic subshifts

In this work, we study the problem of finding the asymptotic growth rate of the number of of $d$-dimensional arrays with side length $n$ over a given alphabet which avoid a list of one-dimensional "forbidden" words along all cardinal directions, as both $n$ and $d$ tend to infinity. Louidor, Marcus, and the second author called this quantity the "limiting entropy"; it is the limit of a sequence of topological entropies of a sequence of isotropic $\mathbb{Z}^d$ subshifts with the dimension $d$ tending to infinity. We find an expression for this limiting entropy which involves only one-dimensional words, which was implicitly conjectured earlier, and given the name "independence entropy." In the case where the list of "forbidden" words is finite, this expression is algorithmically computable and is of the form $\frac{1}{n} \log k$ for $k,n \in \mathbb{N}$. Our proof also characterizes the weak limits (as $d \rightarrow \infty$) of isotropic measures of maximal entropy; any such measure is a Bernoulli extension over some zero entropy factor from an explicitly defined set of measures. We also demonstrate how our results apply to various models previously studied in the literature, in some cases recovering or generalizing known results, but in other cases proving new ones.

math.DS↗

arXiv2013

Shifts of finite type with nearly full entropy

For any fixed alphabet A, the maximum topological entropy of a Z^d subshift with alphabet A is obviously log |A|. We study the class of nearest neighbor Z^d shifts of finite type which have topological entropy very close to this maximum, and show that they have many useful properties. Specifically, we prove that for any d, there exists beta_d such that for any nearest neighbor Z^d shift of finite type X with alphabet A for which log |A| - h(X) < beta_d, X has a unique measure of maximal entropy. Our values of beta_d decay polynomially (like O(d^(-17))), and we prove that the sequence must decay at least polynomially (like d^(-0.25+o(1))). We also show some other desirable properties for such X, for instance that the topological entropy of X is computable and that the unique m.m.e. is isomorphic to a Bernoulli measure. Though there are other sufficient conditions in the literature which guarantee a unique measure of maximal entropy for Z^d shifts of finite type, this is (to our knowledge) the first such condition which makes no reference to the specific adjacency rules of individual letters of the alphabet.

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