Research

My areas of research are dynamical systems, ordinary differential equations, bifurcation theory, potential theory and  representations of groups both in the  local and the global settings.

Papers

• Difféomorphismes de (C2,0) tangents á l’identité qui préservent la fibration de Hopf. C.R. Acad. Sci. Paris Sér. I Math., v. 332, n.11, p. 1011-1014, 2001.
• Holomorphic extensions of formal objects. Annalli Scuola Normale Superiore Pisa Classe Scienze, v. 3, n.4, p. 657-680, 2004.
• Families of diffeomorphismes without periodic curves. The Michigan Mathematical Journal, v. 53, p. 243-256, 2005.
• Formal classification of unipotent parametrized diffeomorphisms. Ergodic Theory & Dynamical Systems (Print), v. 28, p. 1323-1365, 2008.
• Modulus of analytic classification for unfoldings of resonant diffeomorphisms. Moscow Mathematical Journal (Online), v. 8, p. 1-77, 2008.
• Unfoldings of tangent to the identity diffeomorphisms. Astérisque, v. 323, p. 325-370, 2009.
• Non-embeddability of general unipotent diffeomorphisms up to formal conjugacy. Annales de l’Institut Fourier, v. 59, p. 951-975, 2009.
• Topological classification of families of diffeomorphisms without small divisors. Memoirs of the American Mathematical Society, v. 207, serie 975, 166 p., 2010.
• Embedding smooth and formal diffeomorphisms through the Jordan Chevalley decomposition. Journal of Differential Equations (Print), v. 253, p. 3211-3231, 2012.
• Extension of formal conjugations between diffeomorphisms. Bulletin Brazilian Mathematical Society (Impresso), v. 43, p. 247-283, 2012.
• Fixed points for nilpotent actions on the plane and the Cartwright-Littlewood theorem. Mathematische Zeitschrift, v. 279, p. 849-877, 2014. Joint work with Jaime Velasco and Sebastião Firmo.
  
• Derived length of solvable groups of local diffeomorphisms. Mathematische Annalen, v. 358, p. 701-728, 2014. Joint work with Mitchael Martelo.
• Fixed points of nilpotent actions on S2. Ergodic Theory and Dynamical Systems, v. 36, p. 173-197, 2016.
• Finite orbits for nilpotent actions on the torus. Proceedings  of the American Mathematical Society, v. 146, p. 195-208, 2017.  Joint work with Sebastião Firmo.
• Recurrent orbits of subgroups of local complex analytic diffeomorphisms. Mathematische Zeitschrift, v. 285, p. 519-548, 2017.
• Finite dimensional groups of local diffeomorphisms. Israel Journal of Mathematics, v. 227, p. 289-329, 2018.
• The solvable length of groups of local diffeomorphisms. Journal für die reine und angewandte Mathematik (Crelles Journal), v. 752, p. 105-139, 2019.
• Fixed points of nilpotent actions on R2. Astérisque 415, 113-156 (2020). Joint work with  Sebastião Firmo and Patrice Le Calvez.
• Topological rigidity of unfoldings of resonant diffeomorphisms. Annales de l’Institut Fourier, v. 69, n. 3, p 993-1046, 2019.
• The action of a plane singular holomorphic flow on a non-invariant branch. Joint work with Pedro Fortuny Ayuso. Canadian Journal of Mathematics, v. 72, n. 4, 835-866 (2020). DOI: 10.4153/S0008414X19000221
• Global fixed points for nilpotent actions on the torus. Joint work with Sebastião Firmo. Ergodic Theory and Dynamical Systems, v. 40, n. 12, 3339-3367 (2020).
• Stable manifolds of two-dimensional biholomorphisms asymptotic to formal curve. Joint work with Lorena López-Hernanz, Jasmin Raissy and Fernando Sanz-Sánchez.  International Mathematics Research Notices, v. 2021, p. 12847-12887 (2021).
• The local Poincaré problem for irreducible branches. Joint work with José Cano and Pedro Fortuny Ayuso. Revista Matematica Iberoamericana, v. 37, p. 2229-2244 (2020).
• Boundness of intersection numbers for actions by two-dimensional biholomorphisms. J. Inst. Math. Jussieu 21, No. 5, 1677-1700 (2022).
• On the automorphism group of foliations with geometric transverse structure. Joint work with Laurent Meersseman and Marcel Nicolau. Math. Z. 301, No. 2, 1603-1630 (2022).
• Stable manifolds of biholomorphisms in C^n asymptotic to formal curves. Joint work with Lorena López-Hernanz, Fernando Sanz-Sánchez and Liz Vivas. Proc. Lond. Math. Soc. (3) 125, No. 2, 277-317 (2022).
• A fixed point curve theorem for finite orbits local diffeomorphisms. Joint work with Lucivanio Lisboa. Ergodic Theory Dyn. Syst., Volume 43 , Issue 12 , December 2023 , pp. 4138 – 4165
• Complexity of Puiseux solutions of differential and q-difference equations of order and degree one. Joint work with José Cano and Pedro Fortuny Ayuso. Accepted for publication in Publicacions Matemàtiques.
• The Poincaré problem for reducible curves. Joint work with Pedro Fortuny Ayuso. Accepted for publication in Revista Matematica Iberoamericana.

Preprints

• Multisummability of unfoldings of tangent to the identity diffeomorphisms.
• Fixed points of nilpotent actions on surfaces of negative Euler characteristic.
• Nondegenerate germs of holomorphic foliations with prescribed holonomy. Joint work with Rudy Rosas.

Chapters in books

• Algebraic properties of groups of complex analytic local diffeomorphisms. In VIII Escuela Doctoral Intercontinental de Matemáticas PUCP-UVA 2015.  Editors: Francisco Ugarte and Nuria Corral. Pontifica Universidad Católica de Perú, 2015.
• Description of the Zariski-closure of a group of formal diffeomorphisms. Accepted for publication in Handbook on Foliation Theory. Editors: José Seade and Felipe Cano. Springer.