arXiv subjects

Raimundo Briceño

Publications and source records attributed to Raimundo Briceño.

15 recordsLinked to original sources

arXiv2026

Continuous pointwise ergodicity for semigroup actions on locally compact spaces

We investigate proper actions of arbitrary semigroups on separable locally compact metric spaces, where point orbits are allowed to escape to infinity. An action is pointwise uniquely ergodic when every compact orbit closure supports exactly one invariant probability measure and non-compact orbit closures support none. The associated ergodic map therefore assigns the selected probability measure to non-escaping points and the zero subprobability to escaping ones. Under the hypothesis that compact orbit closures admit at least one invariant measure, we establish that the weak* continuity of this ergodic map together with a vanishing at infinity condition is equivalent to the mean ergodicity of the Koopman representation on the space of continuous functions vanishing at infinity. In consequence, every such function and every finite signed measure split uniquely into invariant components and limits of coboundaries. The corresponding projections are obtained by integration against the ergodic map. Because this operator-theoretic characterization avoids explicit averaging schemes, it remains applicable even to semigroups without Følner sequences. When restricted to countable, discrete, bicancellative, and left amenable semigroups, these properties are shown to be equivalent to the uniform convergence of Følner averages and the weak-star continuity of their dual limits, extending classical results for group actions on compact spaces. Furthermore, we identify the space of ergodic measures with a compactified ergodic quotient, prove that the invariant measure simplex is Bauer, and show that these structural properties descend through proper factor maps. The theoretical framework is complemented by dynamical examples, including a continuously pointwise ergodic subshift that exhibits discontinuous entropy along the ergodic map.

math.DS↗

arXiv2025

Additive realizations of asymptotically additive set maps

Given a countable discrete amenable group, we study conditions under which a set map into a Banach space (or more generally, a complete semi-normed space) can be realized as the ergodic sum of a vector under a group representation, such that the realization is asymptotically indistinguishable from the original map. We show that for uniformly bounded group representations, this property is characterized by the class of bounded asymptotically additive set maps, extending previous work for sequences in Banach spaces and on the case of a single non-expansive linear map. Additionally, we develop a relative version of this characterization, identifying when the additive realization can be chosen within a prescribed target set. As an application, our results generalize central aspects of thermodynamic formalism, bridging the additive and asymptotically additive frameworks.

math.DS↗

arXiv2025

Natural extensions of embeddable semigroup actions

Semigroup actions and their invertible extensions are discussed. First, we develop a theory of natural extensions for continuous actions of countable, embeddable semigroups. Second, we demonstrate that not every surjective such action of a semigroup, which embeds into a group and generates it, can be extended to an action of said group, and that this phenomenon is specific to non-reversible semigroups. Furthermore, we characterize the free group on a semigroup (the group together with the embedding) as the unique pair that always admits such an extension, showing that both the choice of the receiving group and the embedding are crucial for this construction. Next, we prove that the classical notion of a natural extension -- requiring all other invertible extensions to factor through it -- only works in the context of compact extensions of left reversible semigroup actions and fails outside of it, thus providing a characterization of left reversibility. We finish by briefly studying topological dynamical properties of the natural extension in the amenable case.

math.DS↗

arXiv2025

Extensibility and denseness of periodic semigroup actions

We study periodic points and finitely supported invariant measures for continuous semigroup actions. Introducing suitable notions of periodicity in both topological and measure-theoretical contexts, we analyze the space of invariant Borel probability measures associated with these actions. For embeddable semigroups, we establish a direct relationship between the extensibility of invariant measures to the free group on the semigroup and the denseness of finitely supported invariant measures. Applying this framework to shift actions on the full shift, we prove that finitely supported invariant measures are dense for every left amenable semigroup that is residually a finite group and for every finite-rank free semigroup.

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arXiv2025

Ergodic theorems for set maps under weak forms of additivity

We investigate various relaxations of additivity for set maps into Banach spaces in the context of representations of amenable groups. Specifically, we establish conditions under which asymptotically additive and almost additive set maps are equivalent. For Banach lattices, we further show that these notions are related to a third weak form of additivity adapted to the order structure of the space. By utilizing these equivalences and reducing non-additive settings to the additive one by finding suitable additive realizations, we derive new non-additive ergodic theorems for amenable group representations into Banach spaces and streamline proofs of existing results in certain cases.

math.DS↗

arXiv2023

Counting independent sets in amenable groups

Given a locally finite graph $Γ$, an amenable subgroup $G$ of graph automorphisms acting freely and almost transitively on its vertices, and a $G$-invariant activity function $λ$, consider the free energy $f_G(Γ,λ)$ of the hardcore model defined on the set of independent sets in $Γ$ weighted by $λ$. Under the assumption that $G$ is finitely generated and its word problem can be solved in exponential time, we define suitable ensembles of hardcore models and prove the following: if $\|λ\|_\infty < λ_c(Δ)$, there exists a randomized $ε$-additive approximation scheme for $f_G(Γ,λ)$ that runs in time $\mathrm{poly}((1+ε^{-1})\lvert Γ/G \rvert)$, where $λ_c(Δ)$ denotes the critical activity on the $Δ$-regular tree. In addition, if $G$ has a finite index linearly ordered subgroup such that its algebraic past can be decided in exponential time, we show that the algorithm can be chosen to be deterministic. On the other hand, we observe that if $\|λ\|_\infty > λ_c(Δ)$, there is no efficient approximation scheme, unless $\mathrm{NP} = \mathrm{RP}$. This recovers the computational phase transition for the partition function of the hardcore model on finite graphs and provides an extension to the infinite setting. As an application in symbolic dynamics, we use these results to develop efficient approximation algorithms for the topological entropy of subshifts of finite type with enough safe symbols, we obtain a representation formula of pressure in terms of random trees of self-avoiding walks, and we provide new conditions for the uniqueness of the measure of maximal entropy based on the connective constant of a particular associated graph.

math.PR↗

arXiv2023

Thermodynamic formalism for amenable groups and countable state spaces

Given the full shift over a countable state space on a countable amenable group, we develop its thermodynamic formalism. First, we introduce the concept of pressure and, using tiling techniques, prove its existence and further properties such as an infimum rule. Next, we extend the definitions of different notions of Gibbs measures and prove their existence and equivalence, given some regularity and normalization criteria on the potential. Finally, we provide a family of potentials that non-trivially satisfy the conditions for having this equivalence and a non-empty range of inverse temperatures where uniqueness holds.

math.DS↗

arXiv2021

Kieffer-Pinsker type formulas for Gibbs measures on sofic groups

Given a countable sofic group $Γ$, a finite alphabet $A$, a subshift $X \subseteq A^Γ$, and a potential $ϕ: X \to \mathbb{R}$, we give sufficient conditions on $X$ and $ϕ$ for expressing, in the uniqueness regime, the sofic entropy of the associated Gibbs measure $μ$ as the limit of the Shannon entropies of some suitable finite systems approximating $Γ\curvearrowright (X,μ)$. Next, we prove that if $μ$ satisfies strong spatial mixing, then the sofic pressure admits a formula in terms of the integral of a random information function with respect to any $Γ$-invariant Borel probability measure with nonnegative sofic entropy. As a consequence of our results, we provide sufficient conditions on $X$ and $ϕ$ for having independence of the sofic approximation for sofic pressure and sofic entropy, and for having locality of pressure in some relevant families of systems, among other applications. These results complement and unify those of Marcus and Pavlov (2015), Alpeev (2017), and Austin and Podder (2018).

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arXiv2020

An SMB approach for pressure representation in amenable virtually orderable groups

Given a countable discrete amenable virtually orderable group $G$ acting by translations on a $G$-subshift $X \subseteq S^G$ and an absolutely summable potential $Φ$, we present a set of conditions to obtain a special integral representation of pressure $P(Φ)$. The approach is based on a Shannon-McMillan-Breiman (SMB) type theorem for Gibbs measures due to Gurevich-Tempelman (2007), and generalizes results from Gamarnik-Katz (2009), Helvik-Lindgren (2014), and Marcus-Pavlov (2015) by extending the setting to other groups besides $\mathbb{Z}^d$, by relaxing the assumptions on $X$ and $Φ$, and by using sufficient convergence conditions in a mean --instead of a uniform-- sense. Under the fairly general context proposed here, these same conditions turn out to be also necessary.

math.DS↗

arXiv2020

Dismantlability, connectedness, and mixing in relational structures

The Constraint Satisfaction Problem (CSP) and its counting counterpart appears under different guises in many areas of mathematics, computer science, and elsewhere. Its structural and algorithmic properties have demonstrated to play a crucial role in many of those applications. For instance, in the decision CSPs, structural properties of the relational structures involved---like, for example, dismantlability---and their logical characterizations have been instrumental for determining the complexity and other properties of the problem. Topological properties of the solution set such as connectedness are related to the hardness of CSPs over random structures. Additionally, in approximate counting and statistical physics, where CSPs emerge in the form of spin systems, mixing properties and the uniqueness of Gibbs measures have been heavily exploited for approximating partition functions and free energy. In spite of the great diversity of those features, there are some eerie similarities between them. These were observed and made more precise in the case of graph homomorphisms by Brightwell and Winkler, who showed that dismantlability of the target graph, connectedness of the set of homomorphisms, and good mixing properties of the corresponding spin system are all equivalent. In this paper we go a step further and demonstrate similar connections for arbitrary CSPs. This requires much deeper understanding of dismantling and the structure of the solution space in the case of relational structures, and new refined concepts of mixing introduced by Briceño. In addition, we develop properties related to the study of valid extensions of a given partially defined homomorphism, an approach that turns out to be novel even in the graph case. We also add to the mix the combinatorial property of finite duality and its logic counterpart, FO-definability, studied by Larose, Loten, and Tardif.

math.CO↗

arXiv2019

Mixing properties of colorings of the $\mathbb{Z}^d$ lattice

We study and classify proper $q$-colorings of the $\mathbb Z^d$ lattice, identifying three regimes where different combinatorial behavior holds: (1) When $q\le d+1$, there exist frozen colorings, that is, proper $q$-colorings of $\mathbb Z^d$ which cannot be modified on any finite subset. (2) We prove a strong list-coloring property which implies that, when $q\ge d+2$, any proper $q$-coloring of the boundary of a box of side length $n \ge d+2$ can be extended to a proper $q$-coloring of the entire box. (3) When $q\geq 2d+1$, the latter holds for any $n \ge 1$. Consequently, we classify the space of proper $q$-colorings of the $\mathbb Z^d$ lattice by their mixing properties.

math.CO↗

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↗

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

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↗

arXiv2014

The topological strong spatial mixing property and new conditions for pressure approximation

In the context of stationary $\mathbb{Z}^d$ nearest-neighbour Gibbs measures $μ$ satisfying strong spatial mixing, we present a new combinatorial condition (the topological strong spatial mixing property (TSSM)) on the support of $μ$ sufficient for having an efficient approximation algorithm for topological pressure. We establish many useful properties of TSSM for studying strong spatial mixing on systems with hard constraints. We also show that TSSM is, in fact, necessary for strong spatial mixing to hold at high rate. Part of this work is an extension of results obtained by D. Gamarnik and D. Katz (2009), and B. Marcus and R. Pavlov (2013), who gave a special representation of topological pressure in terms of conditional probabilities.

math.DS↗