Stefano Luzzatto

External Fellow

My research area is Dynamical Systems and Ergodic Theory and, more specifically, the study of Statistical Properties of Deterministic Dynamical Systems. The general approach goes back to Birkhoff, Kintchin, Hopf in the 1930’s, and follows a path laid out by Anosov, Sinai, Pesin in the 1960’s-1970’s and further motivated by conjectures and results of Palis, Viana and many other researchers since the 1980’s. The key concept in the area is that of hyperbolicity, of which there are many variations (e.g. uniform, non-uniform, partial) and which roughly speaking means that a dynamical systems exhibits some exponential contraction and/or expansion features. Most of my work revolves around the two key conjectures in the area which can be informally and roughly stated as: 1) most dynamical systems are hyperbolic; and 2) hyperbolic dynamical systems satisfy some non-trivial statistical and ergodic properties, such as the existence of “physical measures”.

Website: www.ictp.it/~luzzatto

Positions Held

I completed my BA in Mathematics at the University of Warwick in 1987, followed by an internship at Marlboro College, Vermont, USA, where I taught remedial mathematics and in 1990 completed an MSc in “Philosophy of Mathematics Education”. In 1995 I obtained my PhD in Dynamical Systems at the International School of Advanced Studies, SISSA, Trieste, Italy and the Instituto de Matematica Pura e Aplicada, IMPA, Rio de Janeiro, Brazil under the supervision of Jacob Palis and Marcelo Viana.

After a few postdoctoral positions and temporary lectureships at Paris VI, Warwick, and UMIST, in 2000 I took up a tenure-track position at Imperial College London where I was subsequently promoted to Reader. In 2009 I moved to my current position as Research Scientist at the Abdus Salam International Centre for Theoretical Physics, ICTP, Trieste, Italy.

I am founder and scientific coordinator of the International Masters of Mathematics capacity building programme for developing countries.

Publications

  • Integrability of Continuous Bundles. Journal for die Reine und Angewandte Mathematik 752 (2019) 229–264 (with S. Tureli and K. War).
  • SRB measures for partially hyperbolic systems whose central direction is weakly expanding. Journal of the European Mathematical Society  19 (2017) 10, 2911-2946 (with J. F. Alves, C.L. Dias, V. Pinheiro)
  • The geometric approach for constructing Sinai-Ruelle-Bowen measures. Journal of Statistical Physics 27 (2017) (With V. Climenhaga and Y. Pesin)
  • A Frobenius Theorem for continuous corank-1 continuous distributions in dimensions 2 and 3. International Journal of Mathematics 27 (2016) 467-493 (with S. Tureli and K. War).
  • Geometry of expanding measures and the liftability problem. Ann. Inst. H Poincaré Anal. Non. Lin.  30 (2013) 101-120 (with J. F. Alves, C. Dias).
  • Statistical properties and decay of correlations for interval maps with critical points and singularities. Communications in Mathematical Physics  320 (2013) 21-35 (with I. Melbourne)
  • Uniform hyperbolic approximation of measures with non-zero Lyapunov exponents. Proceedings of the AMS 141 (2013) 3157-3169. (with F. Sanchez-Salas).
  • Finite resolution dynamics. Foundations of Computational Mathematics, Vol 11, No. 2 (2011) 211-239. (with , P. Pilarczyk.)
  • From rates of mixing to recurrence times via large deviations. Advances in Mathematics, 228 (2011) 1203-1236. (with J.F. Alves, J.M. Freitas, S. Vaienti)
  • Invariant measures for interval maps with critical points and singularities. Advances in Mathematics pp. 1428-1444 (2009) (with V. Araujo, M. Viana.)
  • Quantitative hyperbolicity estimates in one-dimensional dynamics, Nonlinearity 21 (2008) 1967-1987 (with S. Day, H. Kokubu, K. Mischaikow, H. Oka, P. Pilarczyk).
  • Hyperbolicity estimates for random maps with positive Lyapunov exponents. Proceedings of the AMS 136 (2008) 3591-3600 (with Y. Cao, I. Rios).
  • Computable starting conditions for the existence of nonuniform hyperbolicity in one-dimensional maps. Nonlinearity 19, pp. 1657-1695 (2006). (with H. Takahashi.)
  • Topological invariance of the sign of the Lyapunov exponents in one-dimensional maps. Proceedings of the AMS, 134 (2006), no. 1, pp. 265-272. (with H. Bruin).
  • Stable manifolds under very weak hyperbolicity conditions. Journal of Differential Equations, 221, pp. 444-469 (2006) (with M. Holland).
  • Topological invariance of generic non-uniformly expanding one-dimensional maps. Mathematical Research Letters, 13, pp. 343-357 (2006). (with L. Wang).
  • Markov structures and decay of correlations for non-uniformly expanding maps. Ann. de l’Inst. Henri Poincare,  Anal. Non-Lin., 22 (2005), no. 6, pp. 817-839. (with J. F. Alves, V. Pinheiro).
  • The Lorenz attractor is mixing. Communication in Mathematical Physics, 260 (2005), pp. 393-401. (with I. Melbourne, F. Paccaut).
  • Lyapunov exponents and rates of mixing for one-dimensional maps. Ergodic Theory and Dynamical Systems 24 (2004) pp. 637—657  (with J. F. Alves, V. Pinheiro).
  • Rates of decay of correlations for one-dimensional dynamics. Annales Scientifiques de l’École Normale Supérieure, 36 (2003), 621-646. (with H. Bruin, S. van Strien).
  • Parameter exclusions in Hénon-like systems. Russian Mathematical Surveys. 58 (2003) pp. 1053-1092  (with M. Viana)
  • Positive Lyapunov exponents for a family of Lorenz-like maps with criticalities, Astérisque, 261 (2000), 201-237. (with M. Viana).
  • Non-uniformly expanding dynamics in maps with criticalities and singularities. Publ. Math. IHES, 89 (1999), 179 – 226. (with W. Tucker).