## 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.

**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},

}

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

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.

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

Title = {The Lanford--Ruelle theorem for actions of sofic groups},

Year = {2021},

Eprint = {arXiv:2112.02334},

}

**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},

}

**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.*