Homepage of Paulo Varandas

My research interests are in Dynamical Systems and Ergodic Theory. These include uniform and non-uniform hyperbolicity, structural stability, centralizers, thermodynamical formalism, decay of correlations, Poincaré recurrence, large deviations and topological methods in ergodic theory.


Email: paulo.varandas at ufba.br 
            pcvarand at gmail.com


"He who knows all the answers
has not been asked all the questions"


"Para ser grande, sê inteiro: nada          
Teu exagera ou exclui.                          
Sê todo em cada coisa. Põe quanto és
No mínimo que fazes.                           
Assim em cada lago a lua toda             
  Brilha, porque alta vive"                         
 - Antologia de Ricardo Reis -
Fernando Pessoa

“Imagination is more important than knowledge.
For knowledge
is limited to all we now know and
understand, while imagination embraces the entire
world, and all there ever will be to know and understand.”
A. Einstein


Curriculum Lattes

    AMS Math Genealogy Articles & Preprints  Students  Teaching  Miscelanea

Articles & Preprints


 P. Varandas, Correlation decay and recurrence asymptotics for some robust nonuniformly hyperbolic maps,  Journal of Statistical Physics, 133, 813-839, 2008. Preprint version

 P. Varandas, Entropy and Poincaré recurrence from a geometrical viewpoint,  Nonlinearity, 22, 2365-2375 2009. Preprint version

 P. Varandas and M. Viana, Existence, uniqueness and stability of equilibrium states for non-uniformly expanding maps  Annalles de l’Institut Henri Poincaré - Analyse Non-Linéaire, 27, 555-593, 2010. Preprint version

 M. Bessa and P. Varandas, On the entropy of incompressible three-dimensional flows, Qualitative Theory of Dynamical Systems, 10, 11-22, 2011. Preprint version

 P. Varandas, Non-uniform specification and large deviations for weak Gibbs measures, Journal of Statistical Physics, 146, 330-358, 2012. Preprint version

 V. Araújo and P. Varandas, Robust exponential decay of correlations for singular-flows, Communications in Mathematical Physics, 311, 215-246, (2012) & Errata to "Robust exponential decay of correlations for singular-flows" Communications in Mathematical Physics 341:2, 729-731 (2016) Preprint version

 J. Rousseau, P. Varandas and Y. Zhao, Entropy formulas for dynamical systems with mistakes, Discrete and Continuous Dynamical Systems - A, 32:12,4391-4407,(2012). Preprint version

 A. Castro and P. Varandas, Equilibrium states for non-uniformly expanding maps: decay of correlations and strong stability, Annalles de l’Institut Henri Poincaré - Analyse Non-Linéaire, 30:2 (2013) 225-249. Preprint version

 P. Varandas and Y. ZhaoWeak specification properties and large deviations for non-additive potentials, Ergodic Theory and Dynamical Systems, 35:3 (2015) 968-993. Preprint version

 P. Varandas, Statistical properties of generalized Viana maps, Dynamical Systems: An International Journal, 29:2 (2014) 167-189. Preprint version

 J. Rousseau, P. Varandas and B. Saussol) Exponential law for random subshifts of finite type, Stochastic Processes and their Applications, 124 (2014) 3260-3276. Preprint version

 C. Lizana, V. Pinheiro and P. Varandas, Contribution to the ergodic theory of robustly transitive maps, Discrete and Continuous Dynamical Systems - A, .35 (2015) 353-365. Preprint version

 M. Bessa and P. Varandas, Trivial and simple spectrum for SL(2,R) cocycles with free base and fiber dynamics, Acta Mathematica Sinica, 31:7 (2015) 1113-1121.
Preprint version

 N. Sumi, P. Varandas and K. Yamamoto, Partial hyperbolicity and specification, Proceedings of the American Mathematical Society, 144:3 (2016) 1161–1170. Preprint version

 P. Varandas, A version of Kac's lemma for suspension flows, Stochastics and Dynamics 16:2, (2106) Preprint version

  I. Melbourne and P. Varandas, A note on statistical properties for nonuniformly hyperbolic systems with slow contraction and expansion, Stochastics and Dynamics, 16:3 (2016) Preprint version

 V. Araújo, I. Melbourne and P. Varandas, Rapid mixing for the Lorenz attractor and statistical limit laws for their time-1 map, Communications in Mathematical Physics 340 (2015) 901–938. Preprint version

 N. Sumi, P. Varandas and K. Yamamoto, Specification and partial hyperbolicity for flows, Dynamical Systems: An International Journal, 30:4, (2015) 501-524. Preprint version

 T. Bomfim and P. Varandas, Multifractal analysis of the irregular set for almost-additive sequences via large deviations, Nonlinearity, 28 (2015) 3563-3585. Preprint version

 M. Carvalho and P. Varandas, (Semi)continuity of the entropy of Sinai probability measures for partially hyperbolic diffeomorphisms, Journal of Mathematical Analysis and Applications, 434:2 (2016) 1123-1137. Preprint version

 V. Araújo, O. Butterley and P. Varandas, Open sets of Axiom A flows with exponentially mixing attractors, Proceedings of the American Mathematical Society 144 (2016) 2971-2984. Preprint version

Proc. Amer. Math. Soc. 144 (2016), 2971-2984Preprint version

  P. Varandas and Y. Zhao, Weak Gibbs measures: convergence to entropy, topological and geometrical aspects, Ergodic Theory and Dynamical Systems
http://dx.doi.org/10.1017/etds.2016.14 (to appear) Preprint version

 T. Bomfim, A. Castro and P. Varandas,  Differentiability of thermodynamical quantities, Advances in Mathematics, 292 (2016), 478–528. Preprint version

 F. Rodrigues and P. Varandas, Specification properties for group actions and thermodynamics of expanding semigroups, Journal of Mathematical Physics 57, (2016) 052704  Preprint version

➣ M. CarvalhoF. Rodrigues and P. Varandas, Semigroups actions of expanding maps, Journal of Statistical Physics 116: 1 (2017), 114-136. Preprint ArXiv:1601.04275

 M. Bessa, C. Ferreira, J. Rocha and P. Varandas, Generic Hamiltonian dynamics, Journal of Dynamics and Differential Equations, 29:1 (2017) 203-2018.  Preprint version

T. Bomfim, M. J. Torres and P. Varandas, Topological features of flows with the reparametrized gluing orbit property, Journal of Differential Equations, 262:8 (2017) 4292-4313 Preprint version

M. BessaJ. Bochi, M. Cambraínha, C. Matheus, P. Varandas and D. Xu, Positivity of top Lyapunov exponent for cocycles on semisimple Lie groups over hyperbolic bases, Buletin of the Brazilian Mathematical Society, DOI: 10.1007/s00574-017-0048-6 Preprint version

X. Tian and P. Varandas, Topological entropy of level sets of empirical measures for non-uniformly expanding maps, Discrete and Continuous Dynamical Systems - Series A, 37:10 (2017) 5407-5431 Preprint version

 T. Bomfim and P. Varandas, Multifractal analysis for weak Gibbs measures: from large deviations to irregular sets, Ergodic Theory and Dynamical Systems, 37: 1 (2017) 79-102 Preprint version

➣ J. Rocha and P. Varandas, The centralizer of C^r generic diffeomorphisms at hyperbolic basic sets is trivial, Proceedings of the American Mathematical Society, 146:1 (2018) 247-260 Preprint version

J. Rocha and P. Varandas, On sensitivity to initial conditions and uniqueness of conjugacies for structurally stable diffeomorphisms, Nonlinearity (to appear) Preprint version

W. Bonomo and J. Rocha and P. Varandas, The centralizer of Komuro-expansive flows and expansive Rd actions, Mathematische Zeitschrift (to appear) Preprint version

➣ M. CarvalhoF. Rodrigues and P. Varandas, Quantitative Poincaré recurrence for semigroup actions, Nonlinearity (to appear) Preprint version



M. Bessa and J. Rocha and P. Varandas, Uniform hyperbolicity revisited: periodic points and equidimensional cycles, Preprint version

T. Bomfim and P. Varandas, The orbit gluing property and large deviation principles for semiflows, Preprint ArXiv:1507.03905

➣ M. Bessa, M. J. Torres and P. Varandas, On the periodic orbits, shadowing and strong transitivity of continuous flows, Preprint version

L. Backes, M. Poletti and P. Varandas, Simplicity of Lyapunov spectrum for linear cocycles over non-uniformly hyperbolic systems, Preprint version

➣ A. Cruz and P. Varandas, SRB measures for partially hyperbolic attractors of endomorphisms, Preprint version

➣ A. Castro, F. Rodrigues and P. Varandas, Stability and limit theorems for sequences of uniformly hyperbolic dynamics, Preprint version

W. Bonomo and P. Varandas, A criterium for the triviality of the centralizer for flows and applications, Preprint version




➣  P. Varandas, Topological methods in dynamics with applications in ergodic theory: old and new, CIM

➣  P. Varandas, PhD Thesis - IMPA 




[1] Roberto Ribeiro Júnior - Existence and uniqueness of weak solution for a hyperbolic parabolic nonlinear partial differential equation, 2009

[2] Kátia Silene Lima Rocha - Singular-hyperbolic attractors are chaotic, 2011

[3] Andrêssa Lima de Souza - Non-uniformly hyperbolic attractors, 2012

[4] David Pinto Martins - Sequencias, progressões e séries: uma abordagem elementar para o ensino médio (PROFMAT), 2013.

[5] José Roberto Teixeira de Campos. Sistemas de equações lineares: uma abordagem geométrica no ensino médio (PROFMAT), 2013.

[6] Marcus Vinícius da Conceição Morro - Specification property and hyperbolicity, 2013

[7] Anderson Reis da Cruz - Ergodic properties of dynamical systems with specification, 2013

[8] Caroline Morais Batista - Topological entropy and homoclinic tangencies, 2014



[1] Wescley Bonomo,  Contributions for the study of the centralizer of flows, Hamiltonians and IR^d actions (2013 - 2016)

[2] Anderson Reis da Cruz, Existence, uniqueness and stability of SRB measures for non-uniformly hyperbolic endomorphisms (2013 - 2016)

[3] Marcus Vinícius da Conceição Morro, (2013 - )

[4] Heides Lima de Santana, (2015 - )

[5] Diego Daltro Conceição, (2015 - )



Álgebra Linear


Dinâmica Hiperbólica

Equações Diferenciais Ordinárias

Fundamentos de Matemática Elementar I-A

Tópicos de Sistemas Dinâmicos

Seminários em Sistemas Dinâmicos



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