Felipe Arbulú

Laboratoire Amiénois de Mathématique Fondamental et Appliquée

Université de Picardie Jules Verne

photo

Publications

  1. with F. Durand and B. Espinoza.
    "The Jacobs–Keane theorem from the \(\mathcal{S}\)-adic viewpoint"
    Discrete and Continuous Dynamical Systems 44. 10 (2024) pp. 3077–3108
    . Journal. Postprint. .
    In the light of recent developments of the \(\mathcal{S}\)-adic study of subshifts, we revisit, within this framework, a well-known result on Toeplitz subshifts due to Jacobs–Keane giving a sufficient combinatorial condition to ensure discrete spectrum. We show that the notion of coincidences, originally introduced in the '70s for the study of the discrete spectrum of substitution subshifts, together with the \(\mathcal{S}\)-adic structure of the subshift allow to go deeper in the study of Toeplitz subshifts. We characterize spectral properties of the factor maps onto the maximal equicontinuous topological factors by means of coincidences density. We also provide an easy to check necessary and sufficient condition to ensure unique ergodicity for constant length \(\mathcal{S}\)-adic subshifts. 
    @article{arbuluJacobsKeane24,
 AUTHOR = {Arbulú, Felipe and Durand, Fabien and Espinoza, Bastián},
 TITLE = {The Jacobs–Keane theorem from the $\mathcal{S}$-adic viewpoint}, 
 YEAR = {2024},
 JOURNAL = {Discrete Contin. Dyn. Syst.},
 VOLUME = {44},
 NUMBER = {10},
 PAGES = {3077--3108},
 DOI = {10.3934/dcds.2024052}
}
  2. with F. Durand.
    "Dynamical properties of minimal Ferenczi subshifts"
    Ergodic Theory and Dynamical Systems 43. 12 (2023) pp. 3923–3970
    . Journal. Postprint. .
    We provide an explicit \(\mathcal{S}\)-adic representation of rank-one subshifts with bounded spacers and call the subshifts obtained in this way ‘minimal Ferenczi subshifts’. We aim to show that this approach is very convenient to study the dynamical behavior of rank-one systems. For instance, we compute their topological rank, the strong and the weak orbit equivalence class. We observe that they have an induced system that is a Toeplitz subshift having discrete spectrum. We also characterize continuous and non-continuous eigenvalues of minimal Ferenczi subshifts. 
    @article{arbuluFerencziSubshifts23,
 AUTHOR = {Arbulú, Felipe and Durand, Fabien},
 TITLE = {Dynamical properties of minimal Ferenczi subshifts}, 
 YEAR = {2023},
 JOURNAL = {Ergodic Theory Dynam. Systems},
 VOLUME = {43},
 NUMBER = {12},
 PAGES = {3923--3970},
 DOI = {10.1017/etds.2023.7}
}

Preprints

  • with B. Espinoza and A. Maass.
    "Topological weak mixing for certain linear involutions of hyperelliptic type"
    . Preprint. .
    A linear involution is an injective piecewise isometry defined on a pair of disjoint intervals. They are defined by a combinatorial data given by a generalized permutation and a length vector. As it was done for interval exchange transformations, it is conjectured that, except for some combinatorial data, a typical linear involution is measure-theoretically weakly mixing. In this direction, one may first explore the question of topological weak mixing for topological models of linear involutions. In this article we prove topological weak mixing for the natural symbolic codings of typical linear involutions defined by some generalized permutations of hyperelliptic type. We consider the generalized permutation \[\sigma_{s,r} = \begin{pmatrix} 0 & A & 1 & 2 & \cdots & s & A & s+1 & s+2 & \cdots & s+r \\ s+r & \cdots & s+2 & s+1 & B & s & \cdots & 2 & 1 & B & 0 \end{pmatrix}. \] We prove that the natural symbolic coding of a typical linear involution defined by a generalized permutation in the Rauzy class of \(\sigma_{s, r}\) is topologically weakly mixing, provided that it has at least a simple letter and its associated genus is sufficiently large. 
    @online{arbuluLinearInvolutions26,
 AUTHOR = {Arbulú, Felipe and Espinoza, Bastián and Maass, Alejandro},
 TITLE = {Topological weak mixing for certain linear involutions of hyperelliptic type}, 
 YEAR = {2026},
 EPRINTTYPE = {arxiv},
 EPRINTCLASS = {math.DS},
 EPRINT = {2607.16830}
}

Talks

Conferences

  • "The Jacobs–Keane theorem from the \(\mathcal{S}\)-adic viewpoint"
    Workshop ANR IZES
    2024-06-30, Île de Porquerolles (Porquerolles, France)

Seminars

  • "Dynamical properties of minimal Ferenczi subshifts"
    Séminaire equipe ADA
    2023-02-02, LMPA (Calais, France)
  • "Dynamical properties of minimal Ferenczi subshifts"
    Séminaire Ernest
    2023-01-10, Institut de Mathématiques de Marseille (Marseille, France)