## 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).