arXiv subjects

Thiago Raszeja

Publications and source records attributed to Thiago Raszeja.

5 recordsLinked to original sources

arXiv2026

Subdiagrams and invariant measures for generalized Bratteli diagrams

The results of this paper contribute to the study of invariant measures of Borel dynamical systems that can be modeled using generalized Bratteli diagrams. In this context, we study tail invariant measures on the path spaces of generalized Bratteli diagrams, allowing countably infinite vertex sets at each level. Our main focus is on subdiagrams of generalized Bratteli diagrams and the problem of extending tail invariant probability measures from vertex and edge subdiagrams to the ambient diagram. We establish necessary and sufficient conditions for the finiteness of such extensions, formulated in terms of incidence matrices and associated stochastic matrices. Several classes of generalized Bratteli diagrams and their subdiagrams are analyzed in detail, including simple, stationary, and bounded size diagrams. We develop constructive, step-by-step procedures for measure extension and for approximating invariant measures by measures supported on suitable subdiagrams. In addition, we provide explicit examples of generalized Bratteli diagrams that admit no probability tail invariant measures, a phenomenon absent for standard Bratteli diagrams with finite vertex sets. Finally, we address convergence questions for sequences of invariant measures arising from approximations by subdiagrams, clarifying the relationship between combinatorial structure and measure-theoretic behavior.

math.DS↗

arXiv2025

Extendable Shift Maps and Weighted Endomorphisms on Generalized Countable Markov Shifts

We obtain an operator algebraic characterization for when we can continuously extend the shift map from a standard countable Markov shift $Σ_A$ to its respective generalized countable Markov shift $X_A$ (a compactification of $Σ_A$). When the shift map is continuously extendable, we obtain explicit formulas for the spectral radius of weighted endomorphisms $aα$, where $α$ is dual to the shift map and conjugated to $Θ(f)=f \circ σ$ on $C(X_A)$, extending a theorem of Kwaśniewski and Lebedev from finite to countable alphabets.

math.DS↗

arXiv2023

Gibbs Measures on Multidimensional Spaces. Equivalences and a Groupoid Approach

We consider some of the main notions of Gibbs measures on subshifts introduced by different communities, such as dynamical systems, probability, operator algebras, and mathematical physics. For potentials with $d$-summable variation, we prove that several of the definitions considered by these communities are equivalent. In particular, when the subshift is of finite type (SFT), we show that all definitions coincide. In addition, we introduced a groupoid approach to describe some Gibbs measures, allowing us to show the equivalence between Gibbs measures and KMS states (the quantum analogous to the Gibbs measures).

math-ph↗

arXiv2022

Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States

Given a 0-1 infinite matrix $A$ and its countable Markov shift $Σ_A$, one of the authors and M. Laca have introduced a kind of {\it generalized countable Markov shift} $X_A=Σ_A \cup Y_A$, where $Y_A$ is a special set of finite admissible words. For some of the most studied countable Markov shifts $Σ_A$, $X_A$ is a compactification of $Σ_A$, and always it is at least locally compact. We developed the thermodynamic formalism on the space $X_A$, exploring the connections with standard results on $Σ_A$. New phenomena appear, such as new conformal measures and a {\it length-type phase transition}: the eigenmeasure lives on $Σ_A$ at high temperature and lives on $Y_A$ at low temperature. Using a pressure-point definition proposed by M. Denker and M. Yuri for iterated function systems, we proved that the Gurevich pressure is a natural definition for the pressure function in the generalized setting. For the gauge action, the Gurevich entropy is a critical temperature for the existence of new conformal measures (KMS states) living on $Y_A$. We exhibit examples with infinitely (even uncountable) many new extremal conformal measures, undetectable in the usual formalism. We prove that conformal measures always exist at low temperatures when the potential is coercive enough. We characterized a basis of the topology of $X_A$ to study the weak$^*$ convergence of measures on $X_A$, and we show some cases where the conformal measure living on $Y_A$ converges to a conformal one living on $Σ_A$. We prove the equivalence among several notions of conformality for locally compact Hausdorff second countable spaces, including quasi-invariant measures for generalized Renault-Deaconu groupoids.

math-ph↗

arXiv2021

Thermodynamic Formalism for Generalized Countable Markov Shifts

Countable Markov shifts, denoted by $Σ_A$ for a 0-1 infinite matrix $A$, are central objects in symbolic dynamics and ergodic theory. R. Exel and M. Laca introduced the corresponding operator algebras, a generalization of the Cuntz-Krieger algebras for infinite countable alphabet, and the set $X_A$, a kind of Generalized Markov Shift (GMS) that coincides with $Σ_A$ in the locally compact case. The set $Σ_A$ is dense in $X_A$, and its complement, a set of finite allowed words, is dense in $X_A$ when non-empty. We develop the thermodynamic formalism for $X_A$, introducing the notion of conformal measure in it, and exploring its connections with the usual formalism for $Σ_A$. New phenomena appear, as different types of phase transitions and new conformal measures undetected by the classical thermodynamic formalism for $A$ not row-finite. Given a potential $F$ and inverse of temperature $β$, we study the existence of conformal measures $μ_β$ associated to $βF$. We present examples where there exists a critical $β_c$ s. t. we have existence of conformal probabilities satisfying $μ_β(Σ_A)=0$ for every $β> β_c$ and, on the weak$^*$ topology, the set of conformal probabilities for $β>β_c$ collapses to the standard conformal probability $μ_{β_c}$, $μ_{β_c}(Σ_A)=1$, for the limit $β\toβ_c$. We study in detail the generalized renewal shift and modifications of it. We highlight the bijection between infinite emitters of the alphabet and extremal conformal probabilities for this class of renewal type shifts. We prove the existence and uniqueness of the eigenmeasure probability of the Ruelle's transformation at low enough temperature for a particular potential on the generalized renewal shift; such measures are not detected on the standard renewal shift since for low temperatures, $βF$ is transient.

math-ph↗