## Overview

My research aims to study the rich interplay between the dynamics of group actions and computability to further understand both phenomena from a unified perspective.

A more detailed exposition can be found by clicking on my self portrait on the right.

**Chaos in Bidimensional Models with Short-Range.**

With Rodrigo Bissacot, Gregório Dalle Vedove and Philippe Thieullen.

doi = {10.48550/ARXIV.2208.10346},

url = {https://arxiv.org/abs/2208.10346},

author = {Barbieri, Sebastián and Bissacot, Rodrigo and Vedove, Gregório Dalle and Thieullen, Philippe},

title = {Chaos in Bidimensional Models with Short-Range},

publisher = {arXiv},

year = {2022},

copyright = {arXiv.org perpetual, non-exclusive license},

}

**Indistinguishable asymptotic pairs and multidimensional Sturmian configurations.**

With Sébastien Labbé.

Author = {Sebasti{\'{a}}n Barbieri and S{\'e}bastien Labb{\'e}},

Title = {Indistinguishable asymptotic pairs and multidimensional Sturmian configurations},

Year = {2022},

Eprint = {arXiv:2204.06413},

}

**Groups with self-simulable zero-dimensional dynamics.**

With Mathieu Sablik and Ville Salo.

Author = {Sebasti{\'{a}}n Barbieri and Mathieu Sablik and Ville Salo},

Title = {Groups with self-simulable zero-dimensional dynamics},

Year = {2021},

Eprint = {arXiv:2104.05141},

}

**Aperiodic subshifts of finite type on groups which are not finitely generated.**

*to appear in Proceedings of the American Mathematical Society.*

Author = {Sebasti{\'{a}}n Barbieri},

Title = {Aperiodic subshifts of finite type on groups which are not finitely generated},

Year = {2022},

Eprint = {arXiv:2204.02447},

}

**The Lanford–Ruelle theorem for actions of sofic groups.**

With Tom Meyerovitch.

*Transactions of the American Mathematical Society, 376(2):1299-9947, 2023.*

As applications of our main result we present a criterion for uniqueness of an equilibrium measure, as well as sufficient conditions for having that the equilibrium states do not depend upon the chosen sofic approximation sequence. We also prove that for any group-shift over a sofic group, the Haar measure is the unique measure of maximal sofic entropy for every sofic approximation sequence, as long as the homoclinic group is dense. On the expository side, we present a short proof of Chung's variational principle for sofic topological pressure.

AUTHOR = {Barbieri, Sebasti\'{a}n and Meyerovitch, Tom},

TITLE = {The {L}anford-{R}uelle theorem for actions of sofic groups},

JOURNAL = {Trans. Amer. Math. Soc.},

FJOURNAL = {Transactions of the American Mathematical Society},

VOLUME = {376},

YEAR = {2023},

NUMBER = {2},

PAGES = {1299--1342},

ISSN = {0002-9947},

MRCLASS = {37B10 (37D35 82B20)},

MRNUMBER = {4531676},

DOI = {10.1090/tran/8810},

URL = {https://doi.org/10.1090/tran/8810},

}

**Markovian properties of continuous group actions: algebraic actions, entropy and the homoclinic group.**

With Felipe García Ramos and Hanfeng Li.

*Advances in Mathematics, 397(108196):1–52, 2022.*

doi = {10.1016/j.aim.2022.108196},

url = {https://doi.org/10.1016/j.aim.2022.108196},

year = {2022},

month = mar,

publisher = {Elsevier {BV}},

volume = {397},

pages = {108196},

author = {Sebasti{\'{a}}n Barbieri and Felipe Garc{\'{\i}}a-Ramos and Hanfeng Li},

title = {Markovian properties of continuous group actions: Algebraic actions, entropy and the homoclinic group},

journal = {Advances in Mathematics}

}

**On the entropies of subshifts of finite type on countable amenable groups.**

*Groups, Geometry and Dynamics, 15(2):607–638, 2021.*

doi = {10.4171/ggd/608},

url = {https://doi.org/10.4171/ggd/608},

year = {2021},

month = jul,

volume = {15},

number = {2},

pages = {607--638},

publisher = {European Mathematical Society - {EMS} - Publishing House {GmbH}},

author = {Sebasti{\'{a}}n Barbieri},

title = {On the entropies of subshifts of finite type on countable amenable groups},

journal = {Groups, Geometry, and Dynamics}

}

**A hierarchy of topological systems with completely positive entropy.**

With Felipe García-Ramos.

*Journal d'Analyse Mathématique, 143(2):639-680, 2021.*

Author = {Sebasti{\'{a}}n Barbieri and Felipe Garc{\'{i}}a-Ramos},

doi = {10.1007/s11854-021-0167-2},

volume = {143},

number = {2},

pages = {639--680},

journal = {Journal d{\textquotesingle}Analyse Math{\'{e}}matique}

title = {A hierarchy of topological systems with completely positive entropy},

publisher = {Springer Science and Business Media {LLC}},

url = {https://doi.org/10.1007/s11854-021-0167-2},

year = {2021},

month = jun

}

**A characterization of Sturmian sequences by indistinguishable asymptotic pairs**

With Sébastien Labbé and Štěpán Starosta.

*European Journal of Combinatorics, 95(103318):1-22, 2021.*

title = {A characterization of Sturmian sequences by indistinguishable asymptotic pairs},

journal = {European Journal of Combinatorics},

volume = {95},

pages = {103318},

year = {2021},

issn = {0195-6698},

doi = {https://doi.org/10.1016/j.ejc.2021.103318},

url = {https://www.sciencedirect.com/science/article/pii/S019566982100010X},

author = {Sebastián Barbieri and Sébastien Labbé and Štěpán Starosta},

abstract = {We give a new characterization of biinfinite Sturmian sequences in terms of indistinguishable asymptotic pairs. Two asymptotic sequences on a full Z-shift are indistinguishable if the sets of occurrences of every pattern in each sequence coincide up to a finitely supported permutation. This characterization can be seen as an extension to biinfinite sequences of Pirillo’s theorem which characterizes Christoffel words. Furthermore, we provide a full characterization of indistinguishable asymptotic pairs on arbitrary alphabets using substitutions and biinfinite characteristic Sturmian sequences. The proof is based on the well-known notion of derived sequences.}

}

**Gibbsian representations of continuous specifications: the theorems of Kozlov and Sullivan revisited.**

With Ricardo Gómez, Brian Marcus, Tom Meyerovitch and Siamak Taati.

*Communications in Mathematical Physics, 382(2):1111–1164, 2021.*

doi = {10.1007/s00220-021-03979-2},

url = {https://doi.org/10.1007/s00220-021-03979-2},

year = {2021},

month = feb,

publisher = {Springer Science and Business Media {LLC}},

volume = {382},

number = {2},

pages = {1111--1164},

author = {Sebasti{\'{a}}n Barbieri and Ricardo G{\'{o}}mez and Brian Marcus and Tom Meyerovitch and Siamak Taati},

title = {{G}ibbsian Representations of Continuous Specifications: The Theorems of {K}ozlov and {S}ullivan Revisited},

journal = {Communications in Mathematical Physics}

}

**Equivalence of relative Gibbs and relative equilibrium measures for actions of countable amenable groups.**

With Ricardo Gómez, Brian Marcus and Siamak Taati.

*Nonlinearity, 33(5):2409–2454, 2020.*

doi = {10.1088/1361-6544/ab6a75},

year = 2020,

month = {mar},

publisher = {{IOP} Publishing},

volume = {33},

number = {5},

pages = {2409--2454},

author = {Sebasti{\'{a}}n Barbieri and Ricardo G{\'{o}}mez and Brian Marcus and Siamak Taati},

title = {Equivalence of relative {G}ibbs and relative equilibrium measures for actions of countable amenable groups},

journal = {Nonlinearity}

}

**A geometric simulation theorem on direct products of finitely generated groups.**

*Discrete Analysis, 9:25, 2019.*

title={A geometric simulation theorem on direct products of finitely generated groups},

doi = {10.19086/da.8820},

journal={Discrete Analysis},

author={Sebasti{\'a}n Barbieri},

number={9},

year={2019},

month = jun,

}

**A generalization of the simulation theorem for semidirect products.**

With Mathieu Sablik.

*Ergodic Theory and Dynamical Systems, 39(12):3185–3206, 2019.*

title={A generalization of the simulation theorem for semidirect products},

volume={39},

DOI={10.1017/etds.2018.21},

number={12},

journal={Ergodic Theory and Dynamical Systems},

publisher={Cambridge University Press},

author={Barbieri, Sebasti{\'{a}}n and Sablik, Mathieu},

year={2019},

pages={3185--3206}

}

**Realization of aperiodic subshifts and uniform densities in groups.**

With Nathalie Aubrun and Stéphan Thomassé.

*Groups, Geometry, and Dynamics, 13(1):107–129, 2019.*

doi = {10.4171/ggd/487},

url = {https://doi.org/10.4171/ggd/487},

year = {2019},

publisher = {European Mathematical Publishing House},

volume = {13},

number = {1},

pages = {107--129},

author = {Nathalie Aubrun and Sebasti{\'{a}}n Barbieri and St{\'{e}}phan Thomass{\'{e}}},

title = {Realization of aperiodic subshifts and uniform densities in groups},

journal = {Groups, Geometry, and Dynamics}

}

**A notion of effectiveness for subshifts on finitely generated groups.**

With Nathalie Aubrun and Mathieu Sablik.

*In Theoretical Computer Science, 661:35–55, 2017.*

title = "A notion of effectiveness for subshifts on finitely generated groups",

journal = "Theoretical Computer Science",

volume = "661",

pages = "35 - 55",

year = "2017",

issn = "0304-3975",

doi = "https://doi.org/10.1016/j.tcs.2016.11.033",

url = "http://www.sciencedirect.com/science/article/pii/S0304397516306983",

author = "Nathalie Aubrun and Sebasti{\'a}n Barbieri and Mathieu Sablik",

keywords = "Symbolic dynamics, Turing machines, Word problems, Models of computation"

}

**The domino problem is undecidable on surface groups.**

With Nathalie Aubrun and Etienne Moutot.

*44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019), pages 1–14, 2019.*

author = {Nathalie Aubrun and Sebasti{\'a}n Barbieri and Etienne Moutot},

title = {{The Domino Problem is Undecidable on Surface Groups}},

booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},

pages = {46:1--46:14},

series = {Leibniz International Proceedings in Informatics (LIPIcs)},

ISBN = {978-3-95977-117-7},

ISSN = {1868-8969},

year = {2019},

volume = {138},

editor = {Peter Rossmanith and Pinar Heggernes and Joost-Pieter Katoen},

publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},

address = {Dagstuhl, Germany},

URL = {http://drops.dagstuhl.de/opus/volltexte/2019/10990},

URN = {urn:nbn:de:0030-drops-109900},

doi = {10.4230/LIPIcs.MFCS.2019.46},

annote = {Keywords: tilings, substitutions, SFTs, decidability, domino problem}

}

**The domino problem for self-similar structures.**

With Mathieu Sablik.

*Pursuit of the Universal (CIE 2016), pages 205–214, 2016.*

author="Barbieri, Sebasti{\'a}n and Sablik, Mathieu",

editor="Beckmann, Arnold and Bienvenu, Laurent and Jonoska, Nata{\v{s}}a",

title="The Domino Problem for Self-similar Structures",

booktitle="Pursuit of the Universal",

year="2016",

publisher="Springer International Publishing",

address="Cham",

pages="205--214",

isbn="978-3-319-40189-8"

}

**The group of reversible Turing machines.**

With Jarkko Kari and Ville Salo.

*Cellular Automata and Discrete Complex Systems (AUTOMATA 2016), pages 49–62, 2016.*

author="Barbieri, Sebasti{\'a}n and Kari, Jarkko and Salo, Ville",

editor="Cook, Matthew and Neary, Turlough",

title="The Group of Reversible Turing Machines",

booktitle="Cellular Automata and Discrete Complex Systems",

year="2016",

publisher="Springer International Publishing",

address="Cham",

pages="49--62",

isbn="978-3-319-39300-1"

}

**About the Domino Problem for Subshifts on Groups.**

With Nathalie Aubrun and Emmanuel Jeandel.

*Sequences, Groups, and Number Theory, pages 331–389. Springer International Publishing, 2018.*

address = {Cham},

series = {Trends in {Mathematics}},

title = {About the {Domino} {Problem} for {Subshifts} on {Groups}},

isbn = {978-3-319-69152-7},

language = {en},

urldate = {2019-05-25},

booktitle = {Sequences, {Groups}, and {Number} {Theory}},

publisher = {Springer International Publishing},

author = {Aubrun, Nathalie and Barbieri, Sebasti{\'{a}}n and Jeandel, Emmanuel},

editor = {Berth{\'{e}}, Val{\'{e}}rie and Rigo, Michel},

year = {2018},

doi = {10.1007/978-3-319-69152-7_9},

pages = {331--389},

}

**Shift spaces on groups: computability and dynamics.**

*Theses, Université de Lyon (ENS de Lyon), June 2017.*

TITLE = {{Shift spaces on groups : computability and dynamics}},

AUTHOR = {Barbieri, Sebasti{\'{a}}n},

URL = {https://tel.archives-ouvertes.fr/tel-01563302},

NUMBER = {2017LYSEN021},

SCHOOL = {{Universit{\'e} de Lyon}},

YEAR = {2017},

MONTH = Jun,

KEYWORDS = {Conjugacy invariants ; Group theory ; Simulation theorems ; Symbolic dynamics ; Dynamical systems ; Shift spaces ; Aperiodicity ; Computability ; Dynamique symbolique ; Syst{\`e}mes dynamiques ; Sous-d{\'e}calages ; Aperiodicit{\'e} ; Calculabilit{\'e} ; Th{\'e}or{\`e}mes de simulation ; Th{\'e}orie des groupes ; Invariants de conjugaison},

TYPE = {Theses},

HAL_ID = {tel-01563302},

HAL_VERSION = {v1},

}

**Tilings on different structures: exploration towards two problems.**

*Mémoire Master 2, 2014.*

**Subshifts generados por sustituciones multidimensionales.**

*Memoria ingeniería Universidad de Chile, 2014.*