"The secret of change is to focus all of your energy not on fighting the old, but on building the new" - Socrates
"Develop a passion for learning. If you do, you'll never cease growing" - Anthony J. D'Angelo
"The secret of change is to focus all of your energy not on fighting the old, but on building the new" - Socrates
"Develop a passion for learning. If you do, you'll never cease growing" - Anthony J. D'Angelo
Thermodynamic formalism
Equilibrium States, Linear Response and Phase Transitions
A firt aim of the thermodynamic formalism in dynamical systems is the construction and study of ergodic properties of invariant measures - equilibrium states - that attain the topological complexity of the dynamics. In the case of a constant potential, equilibrium states are maximal entropy measures. A special interest is given to the dependence of equilibrium states on the potential and the dynamical system, and to the existence of phase transitions. After pioneering work of Sinai, Ruelle and Bowen in the sixties and many contributions in the last decades, it is still a challenge is to understand the thermodynamic formalism of dynamical systems with some non-uniform hyperbolicity.
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.
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
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.
T. Bomfim, A. Castro and P. Varandas, Differentiability of thermodynamical quantities, Advances in Mathematics, 292 (2016), 478–528.
A. Cruz and P. Varandas, SRB measures for partially hyperbolic attractors of endomorphisms, Ergodic Theory and Dynamical Systems, 40:6 (2020) 1545 - 1593
M. Stadlbauer, S. Suzuki and P. Varandas, Thermodynamic formalism for random non-uniformly expanding maps, Communications in Mathematical Physics 385 : 1 (2021) 369427
M. Stadlbauer, P. Varandas and X. Zhang, Quenched and annealed equilibrium states for random Ruelle expanding maps and applications, Ergodic Theory and Dynamical Systems (to appear)
A. Lopes, S. Lopes and P. Varandas, Bayes posterior convergence for loss functions via almost additive thermodynamic formalism, Journal of Statistical Physics 186:35 (2022)
A. Biś, M Carvalho, M. Mendes and P. Varandas, A convex analysis approach to entropy functions, variational principles and equilibrium states, Communications in Mathematical Physics 394 (2022) 215–256
J. F. Alves, W. Bahsoun, M. Ruziboev and P. Varandas, Quenched decay of correlations for nonuniformly hyperbolic random maps with an ergodic driving system, Nonlinearity (to appear)
Statistical properties of weak Gibbs measures
Gibbs measures play a key role in the thermodynamic formalism of uniformly hyperbolic dynamical systems, as equilibrium states for regular potentials arise are invariant Gibbs measures. The concept of weak Gibbs measure appeared naturally in dealing with non-uniformly hyperbolic dynamical systems, where the Gibbs property can be obtained at the instants of hyperbolicity. The geometric aspects of weak Gibbs measures allow one to study the velocity of convergence in the Shannon-McMillan-Breiman, their multifractal analysis and relation with large deviations (even in non-additive contexts).
P. Varandas and Y. Zhao, Weak specification properties and large deviations for non-additive potentials, Ergodic Theory and Dynamical Systems, 35:3 (2015) 968-993.
P. Varandas and Y. Zhao, Weak Gibbs measures: convergence to entropy, topological and geometrical aspects, Ergodic Theory and Dynamical Systems, 37:7, (2017), 2313-2336.
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
I. Melbourne and P. Varandas, Convergence to a Lévy process in the Skorohod M1 and M2 topologies for nonuniformly hyperbolic systems, including billiards with cusps, Communications in Mathematical Physics 375 (2020) 653–678
A. Cruz, G. Ferreira and P. Varandas, Volume lemmas and large deviations for partially hyperbolic endomorphisms, Ergodic Theory and Dynamical Systems 41: 1 (2021) 213 - 240
Y. Shi, X. Tian, P. Varandas and X. Wang, On multifractal analysis and large deviations of singular-hyperbolic attractors Nonlinearity (to appear)
Decay of Correlations and Limit Theorems for Flows
A major problem in ergodic theory is to describe the speed of mixing of dynamical systems, which measures an asymptotic independence of invariant probability measures. This problem has shown to be harder for flows than for diffeomorphisms. The time-1 map of a flow is at most partially hyperbolic, with an invariant foliation given by the orbits of the flow. Even though the neutral behavior of the foliation by orbits, the topological and ergodic mixing properties of hyperbolic and non-uniformly hyperbolic flows have much to do with a shear along the orbits. For instance, constant suspension flows over topologically mixing Anosov diffeomorphisms define transitive but not topologically mixing Anosov flows. We gave first examples of flows (singular and non-singular) with robust exponential decay of correlations. The global understanding of the decay of correlations for hyperbolic and non-uniformly hyperbolic flows, including a solution to Bowen-Ruelle's conjecture in full generality, is still incomplete.
V. Araújo and P. Varandas, Robust exponential decay of correlations for singular-flows, Communications in Mathematical Physics, 311, 215-246, (2012)
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.
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.
D. Daltro and P. Varandas, Exponential decay of correlations for Gibbs measures and semiflows over C^1+ piecewise expanding maps, Annales Henri Poincaré, 22:7 (2021), 2137-2159
Topological dynamics
Specification, gluing orbit properties and their consequences
Besides from its intrinsic interest, the specification property introduced by Bowen is a very useful tool in ergodic theory, in the study of large deviations for continuous observables and multifractal analysis, just to mention two applications. However, this property and the gluing orbit property turned out to be rare beyond the uniformly hyperbolic context, even though it is generic for volume preserving homeomorphisms.
P. Varandas, Non-uniform specification and large deviations for weak Gibbs measures, Journal of Statistical Physics, 146, 330-358, 2012.
N. Sumi, P. Varandas and K. Yamamoto, Partial hyperbolicity and specification, Proceedings of the American Mathematical Society, 144:3 (2016) 1161–1170.
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
T. Bomfim and P. Varandas, The gluing orbit property, uniform hyperbolicity and large deviation principles for semiflows, Journal of Differential Equations 267 (2019) 228–266
A. Castro, F. Rodrigues and P. Varandas, Stability and limit theorems for sequences of uniformly hyperbolic dynamics, Journal of Mathematical Analysis and Applications, 480 (2019) 123426
S. Troubetzkoy and P. Varandas, On the role of continuity and expansiveness on LEO and periodic specification properties, CIRM Jean-Morlet Chair Subseries, Lecture Notes in Mathematics 2290, Springer-Verlag
T. Bomfim, M. J. Torres and P. Varandas, The gluing orbit property and partial hyperbolicity, Journal of Differential Equations 272 (2021) 203–221
M. Bessa, M. J. Torres and P. Varandas, Topological aspects of incompressible flows, Journal of Differential Equations 293 (2021) 392-417
Birkhoff and Lyapunov non-typical points
The ergodic theorems ensure that Birkhoff averages are convergent in a full probability measure set, and similarly whenever the Birhoff averages are replaced by quotients of Birkhoff averages or by sub-additive sequences of maps. In the context of smooth conformal expanding maps it is known that set of points which are non-typical is very rich: it is Baire generic, has full topological pressure and has full Hausdorff dimension. As the specification property, used in the classical argument, is a rather strong requirement, other approaches have been used to describe the set of non-typical points for large classes of dynamical systems.
X. Tian and P. Varandas, Topological entropy of level sets of empirical measures for non-uniformly expanding maps, Discrete and Continuous Dynamical Systems - A, 37:10 (2017) 5407-5431
M. Carvalho and P. Varandas, Genericity of historic behavior for maps and flows, Nonlinearity 34:10 (2021) 7030-7044
M. Carvalho, V. Coelho, L. Salgado and P. Varandas, Sensitivity and historic behavior on Baire metric spaces, Ergodic Theory and Dynamical Systems (to appear)
Metric mean dimension
The metric mean dimension, introduced by Lindenstrauss and Weiss, is an entropy-like invariant that describes the complexity of dynamical systems with infinite topological entropy, including generic homeomorphisms or shifts with a compact and uncountable alphabet.
M. Carvalho, F. Rodrigues and P. Varandas, A variational principle for the metric mean dimension of free semigroup actions, Ergodic Theory and Dynamical Systems 42:1 (2022) 65-85
M. Carvalho , F. Rodrigues and P. Varandas, Generic homeomorphisms have full metric mean dimension, Ergodic Theory and Dynamical Systems 42:1 (2022) 40-64
Rotation Theory
The concept of rotation number introduced by Poincaré to study circle homeomorphisms has been much fruitful as a mechanism to detect periodic points or minimal behavior in this one-dimensional context, among other properties. In the case of continuous maps on the torus homotopic to the identity, Misiurewicz and Ziemian introduced several notions of rotation sets, which are convex and seem to replicate the one-dimensional toolbox in the special case of the 2-torus. The study of totation sets for maps on higher dimensional tori and flows requires the use of different techniques and have been much less studied.
H. Lima and P. Varandas, On the rotation sets of generic homeomorphisms on the torus Td, Ergodic Theory and Dynamical Systems 41 (2021) 2983-3022
W. Bonomo, H. Lima and P. Varandas, The rotation sets of most volume preserving homeomorphisms on Td are stable, convex and rational polyhedrons, Israel Journal of Mathematics 243 (2021) 81-102.
P. Varandas, Realization of rotation vectors for volume preserving homeomorphisms of the torus, Topological Methods in Nonlinear Analysis 60 (2022) 441-455
W. Bonomo, H. Lima and P. Varandas, Shape and stability of rotation sets for incompressible flows on the torus Td, Israel Journal of Mathematics (to appear)
Algebraic aspects in dynamical systems
Centralizers in Dynamical Systems
Smale conjecured that typical smooth diffeomorphisms should have trivial symmetries ie, every symmetry is an iterate of the diffeomorphism. The conjecture was proved in special contexts, including C^r circle diffeomorphisms, hyperbolic basic sets for C^r-diffeomorphisms, C^1-generic diffeomorphisms and C^infty uniformly hyperbolic dynamics on tori. The problem has a negative answer for less regular maps. The symmetries of flows and R^d actions need to contemplate their reparameterizations, and their triviality is associated to the existence of non-trivial first integrals. A complete solution to Smale's conjecture is still unavailable.
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
W. Bonomo and J. Rocha and P. Varandas, The centralizer of Komuro-expansive flows and expansive Rd actions, Mathematische Zeitschrift (2017) https://doi.org/10.1007/s00209-017-1988-7
W. Bonomo and P. Varandas, A criterion for the triviality of the centralizer for flows and applications, Journal of Differential Equations 267:3 (2019) 1748-1766
P. Varandas, On the cohomological class of fiber-bunched cocycles on semi simple Lie groups, Revista Matemática Iberoamericana 36:4 (2020)
W. Bonomo and P. Varandas, Continuous flows generate few homeomorphisms, Proceedings of the Edinburgh Mathematical Society, 63:4 (2020) 971-983
J. Rocha and P. Varandas, On C^0 centralizers of Anosov diffeos on the torus: algebraic and topological aspects, Fundamenta Mathematicae 258 (2022), 1-24
Group and Semigroup Actions
Continuous maps acting on compact metric spaces have invariant probability measures, and satisfy a variational principle for the topological pressure. In the context of group actions by continuous maps on a compact metric space, there exist probability measures invariant by all elements in a group action if and only if the group is amenable. Moreover, while stationary measures play a crucial role in the context of group actions, these need not describe the thermodynamic formalism of semigroup actions. For the development of ergodic theory of non-amenable (semi)group actions it seems necessary to understand the suitable notions of invariance and entropy.
F. Rodrigues and P. Varandas, Specification properties for group actions and thermodynamics of expanding semigroups, Journal of Mathematical Physics 57, (2016) 052704
M. Carvalho, F. Rodrigues and P. Varandas, Semigroups actions of expanding maps, Journal of Statistical Physics 116: 1 (2017), 114-136.
M. Carvalho, F. Rodrigues and P. Varandas, A variational principle for free semigroup actions, Advances in Mathematics, 334 (2018) 450-487
M. Carvalho, F. Rodrigues and P. Varandas, Quantitative Poincaré recurrence for semigroup actions, Nonlinearity, 31:3 (2018) 864-886
Some other topics
Uniform and non-uniform hyperbolicity
In the context of linear cocycles, Oseledets celebrated theorem ensures that almost every point has well defined Lyapunov exponents and invariant subspaces. A major question in dynamics is to understand if non-uniform hyperbolicity is abundant. In a recent work we prove that typical linear cocycles on semisimple Lie groups have at least one positive Lyapunov exponent.
M. Bessa and J. Rocha and P. Varandas, Uniform hyperbolicity revisited: periodic points and equidimensional cycles, Dynamical Systems, 33:4 (2018) 691-707
M. Bessa, J. 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, 49:1 (2018) 73-87.
J. Rocha and P. Varandas, On sensitivity to initial conditions and uniqueness of conjugacies for structurally stable diffeomorphisms, Nonlinearity, 31:293 (2018) 293-313
L. Backes, M. Poletti and P. Varandas, Simplicity of Lyapunov spectrum for linear cocycles over non-uniformly hyperbolic systems (with an Appendix by Y. Lima) Ergodic Theory and Dynamical Systems, 40:11 (2020) 2947-2969
P. Varandas, On the cohomological class of fiber-bunched cocycles on semi simple Lie groups, Revista Matemática Iberoamericana 36:4 (2020) 1113-1132
Quantitative Poincaré recurrence
Poincaré recurrence theorem implies the recurrence of almost every point (with respect to any invariant probability measure). Kac's formula provides a first quantitative estimate on the average expected return time to positive measures subsets. A step further is to provide sharper quantitative estimates for recurrence, e.g. proving limit laws for return and/or hitting times.
P. Varandas, Correlation decay and recurrence asymptotics for some robust nonuniformly hyperbolic maps, Journal of Statistical Physics, 133, 813-839, 2008.
P. Varandas, Entropy and Poincaré recurrence from a geometrical viewpoint, Nonlinearity, 22, 2365-2375 2009.
J. Rousseau, P. Varandas and B. Saussol) Exponential law for random subshifts of finite type, Stochastic Processes and their Applications, 124 (2014) 3260-3276
P. Varandas, A version of Kac's lemma for suspension flows, Stochastics and Dynamics 16:2 (2016) 1660002
Ergodic optimization
Ergodic optimization in dynamical systems concerns the description of the non-emtpy set of invariant measures which have largest (or smallest) time averages for Birkhoff or Kingman ergodic theorems, and study of their ergodic properties. The ergodic optimization for non-uniformly hyperbolic dynamical systems is still not completely understood.
M. Morro, R. Sant'Anna and P. Varandas, Ergodic optimization for hyperbolic flows and Lorenz attractors, Annales Henri Poincaré, 21:10 (2020) 3253-3283
T. Bomfim, R. Huo, P. Varandas and Y. Zhao, Typical properties of ergodic optimization for asymptotically additive potentials, Stochastics and Dynamics, 23: 1 (2023) 2250024