José Pedro Gaivão

Assistant Professor

Department of Mathematics
ISEG, Universidade de Lisboa
Rua do Quelhas, 6
1200-781 Lisboa

Email: jpgaivao at
Phone: (+351) 213 925 874 (Ext. 3874)
Office: 511, Quelhas 6

Member of REM, CEMAPRE and of the Lisbon Dynamical Systems Group - LxDS.

Research interests:

Dynamical Systems, Hamiltonian dynamics, piecewise contractions, billiards and exponentially small phenomena.


Licenciatura in Matemática Aplicada à Tecnologia, Universidade do Porto, 2005.
PhD in Mathematics (supervisor: Vassili Gelfreich), University of Warwick, 2010.


  1. Exponentially small splitting of invariant manifolds near a Hamiltonian-Hopf bifurcation
    PhD thesis, University of Warwick, 2010.
  2. Splitting of separatrices for the Hamiltonian-Hopf bifurcation with the Swift-Hohenberg equation as an example
    (with V. Gelfreich)
    Nonlinearity, 24, no.3, p. 677-698, 2011.
  3. Chaos in the square billiard with a modified reflection law
    (with G. Del Magno, J. L. Dias, P. Duarte and D. Pinheiro)
    Chaos, 22, 026106, 2012.
  4. Analytic invariants for the 1:-1 resonance.
    Annales de l'Institut Fourier, vol. 63, no.4, p. 1267-1426, 2013.
  5. SRB measures for polygonal billiards with contracting reflection laws
    (with G. Del Magno, J. L. Dias, P. Duarte and D. Pinheiro)
    Communications in Mathematical Physics, vol. 329, 2, p. 687-723, 2014.
  6. Ergodicity of polygonal slap maps
    (with G. Del Magno, J. L. Dias and P. Duarte)
    Nonlinearity, 27, no.8, p. 1969-1983, 2014.
  7. Dissipative outer billiards: a case study
    (with G. Del Magno and E. Gutkin)
    Dynamical Systems: An International Journal, vol. 30, no.1, p. 45-69, 2015.
  8. Hyperbolic billiards on polytopes with contracting reflection laws
    (with P. Duarte and M. Soufi)
    Discrete and Continuous Dynamical Systems - Series A, vol. 37, no. 6, p. 3079-3109, 2017.
  9. Hyperbolic polygonal billiards with finitely many ergodic SRB measures
    (with G. Del Magno, J. L. Dias, P. Duarte)
    Ergodic Theory and Dynamical Systems, vol. 38, issue 6, p. 2062-2085, 2018.
  10. $\lambda$-stability of periodic bililard orbits
    (with S. Troubetzkoy)
    Nonlinearity, 31, no.9, p. 4326, 2018.
  11. Linearization of Gevrey flows on $\mathbb{T}^d$ with a Brjuno type arithmetical condition
    (with J. L. Dias)
    Journal of Differential Equations, vol. 267, issue 12, p. 7167-7212, 2019.
  12. Asymptotic periodicity in outer billiards with contraction
    Ergodic Theory and Dynamical Systems, vol. 40, issue 2, p. 402-417, 2020.
  13. On the attractor of piecewise expanding maps of the interval
    (with G. Del Magno, J. L. Dias, P. Duarte)
    Stochastics and Dynamics, vol. 20, No. 02, 2050009, 2020.
  14. Periodic attractor in the discrete time best-response dynamics of the Rock-Paper-Scissors game
    (with T. Peixe)
    Dynamic Games and Applications,11(3), 491-511, 2021.
  15. Hyperbolic polygonal billiards close to 1-dimensional piecewise expanding maps
    (with G. Del Magno, J. L. Dias, P. Duarte)
    Journal of Statistical Physics, vol. 182, No. 11, 2021.
  16. Dynamics of piecewise increasing contractions
    (with A. Nogueira)
    Bulletin of the London Mathematical Society, Online First, doi: 10.1112/blms.12577, 2022.
  17. Billiards in generic convex bodies have positive topological entropy
    (with M. Bessa, G. Del Magno, J. L. Dias, M. J. Torres)
    Preprint, 2022.

Last update: 10 January 2022