Rua do Quelhas 6, 1200-781 Lisboa, Portugal
Email:
Tel: +351 213 925 833

Teaching

PhD program DMAEG

Undergraduate degree MAEG (infocursos)

LxDS Lisbon Dynamical Systems group


Research interests in Mathematics

Publications

  1. Renormalisation of vector fields,
    PhD thesis, University of Cambridge, 2002.
  2. Renormalization of flows on the multidimensional torus close to a KT frequency vector,
    Nonlinearity 15, 647-664 (2002).
  3. Renormalization scheme for vector fields on $\mathbb T^2 $with a diophantine frequency,
    Nonlinearity 15, 665-679 (2002).
  4. Brjuno condition and renormalization for Poincaré flows,
    Discrete Contin. Dyn. Syst. 15, 641-656 (2006).
  5. A normal form theorem for Brjuno skew-systems through renormalization,
    J. Differential Equations 230, 1-23 (2006).
  6. Renormalization for multidimensional Hamiltonian flows,
    (with K. Khanin and J. Marklof)
    Nonlinearity 19, 2727-2753 (2006).
  7. Multidimensional continued fractions, dynamic renormalization and KAM theory,
    (with K. Khanin and J. Marklof)
    Commun. Math. Phys. 270, 197-231 (2007).
  8. Renormalization of diophantine skew flows, with applications to the reducibility problem,
    (with H. Koch)
    Discrete Contin. Dyn. Syst. 21, 477-500 (2008).
  9. Generic dynamics of 4-dimensional $C^2$ Hamiltonian systems,
    (with M. Bessa)
    Commun. Math. Phys. 281, 597-619 (2008).
  10. Local conjugacy classes for analytic torus flows,
    J. Differential Equations 245, 468-489 (2008).
  11. Hamiltonian elliptic dynamics on symplectic 4-manifolds,
    (with M. Bessa)
    Proc. Amer. Math. Soc. 137, 585-592 (2009).
  12. On the Herman-Avila-Bochi formula for Lyapunov exponents of $\operatorname{SL}(2,\mathbb R)$-cocycles,
    (with A. T. Baraviera and P. Duarte)
    Nonlinearity 24, 2465-2476 (2011).
  13. Chaos in the square billiard with a modified reflection law,
    (with G. Del Magno, P. Duarte, J. P. Gaivão and D. Pinheiro)
    Chaos 22, 026106 (2012).
  14. Hamiltonian suspension of perturbed Poincaré sections and an application,
    (with M. Bessa)
    Math. Proc. Cam. Phil. Soc. 157, 101-112 (2014).
  15. SRB measures for polygonal billiards with contracting reflection laws,
    (with G. Del Magno, P. Duarte, J. P. Gaivão and D. Pinheiro)
    Commun. Math. Phys. 329, 687-723 (2014).
  16. Realization of tangent perturbations in discrete and continuous time conservative systems,
    (with H. N. Alishah)
    Discrete Contin. Dyn. Syst. 34, 5359-5374 (2014).
  17. Ergodicity of polygonal slap maps,
    (with G. Del Magno, P. Duarte and J. P. Gaivão)
    Nonlinearity 27, 1969-1983 (2014).
  18. On shadowing and hyperbolicity for geodesic flows on surfaces,
    (with M. Bessa and M. J. Torres)
    Nonlinear Analysis 155, 250-263 (2017).
  19. Hyperbolic polygonal billiards with finitely many ergodic SRB measures,
    (with G. Del Magno, P. Duarte and J. P. Gaivão)
    Ergodic Theory and Dynamical Systems (to appear).
  20. Ergodic properties of polygonal billiards with strongly contracting reflection laws,
    (with G. Del Magno, P. Duarte and J. P. Gaivão)
    preprint (2015).
  21. Reducibility of quasi-periodically forced circle flows,
    (with S. Kocic)
    preprint (2016).
  22. Renormalization of Gevrey vector fields with a Brjuno type arithmetical condition,
    (with J. P. Gaivão)
    preprint (2017).

Conference proceedings

  1. KAM theory as a limit of renormalization,
    in Dynamics, Games and Science I, Eds. Peixoto, Pinto and Rand, Springer, 2011.
  2. Generic Hamiltonian dynamical systems: an overview,
    (with M. Bessa)
    in Dynamics, Games and Science I, Eds. Peixoto, Pinto and Rand, Springer, 2011.
  3. From ice to penguins: the role of mathematics in Antarctic research,
    J. C. Xavier, S. L. Hill, M. Belchier, T. Bracegirdle, E. J. Murphy, J. Lopes Dias,
    in Mathematics of Planet Earth. CIM Series in Mathematical Sciences, Eds. Bourguignon, Jeltsch, Pinto and Viana, Springer-Verlag, 2014.
  4. Review on non-perturbative reducibility of quasi-periodically forced linear flows with two frequencies,
    in Dynamics, Games and Science. CIM Series in Mathematical Sciences, Eds. Bourguignon, Jeltsch, Pinto and Viana, Springer-Verlag, 253-271, 2014.
  5. Polygonal billiards with strongly contractive reflection laws: a review of some hyperbolic properties,
    (with G. Del Magno, P. Duarte, J. P. Gaivão and D. Pinheiro)
    in Difference Equations, Discrete Dynamical Systems and Applications, Eds. L. Alseda i Soler et al., Springer Proceedings in Mathematics and Statistics 180, 179-190, 2016.

Books

  1. Introdução à Integração em Rn, Coleção Económicas - 2.a Série, Almedina, 2015.

Lecture Notes, etc.

  1. Difeomorfismo minimal com entropia topológica não nula, 1997 [in portuguese].
  2. Notes on the translated curve theorem, 1998.
  3. Dynamics of Differential Equations (Math Tripos IIA), random exercises for supervisions, Lent 1998.
  4. Fast notes on symplectomorphisms, 2001.
  5. Multidimensional continued fractions and renormalization, Renormalization in Dynamical Systems Workshop, Fields Institute, Toronto, 2005.
  6. Notas de Análise Complexa, 2005 [in portuguese].
  7. Notas de Análise de Fourier e aplicações a edp's, 2005 [in portuguese].
  8. Notas de Análise Matemática I, 2006 [in portuguese].
  9. Notas de Análise Numérica, 2008 [in portuguese].
  10. Notas de Análise Matemática III, 2009 [in portuguese].
  11. An interview with Robert MacKay, CIM Bulletin 31, 15-18, 2012.
  12. Mathematical Economics: Optimization, 2015

Education

Supervision

Postdocs
PhD
MSc
Undergraduate

Dynamic website development

Links, etc.


Lisbon, Portugal


By a stone we lost a horseshoe.
By a horseshoe we lost a knight.
By a knight we lost a battle.
By a battle we lost the kingdom.

i.e. sensitivity to the initial conditions