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Probabilistic and algebraic aspects of smooth dynamical systems

Grant number: 17/06463-3
Support Opportunities:Research Projects - Thematic Grants
Duration: February 01, 2018 - January 31, 2025
Field of knowledge:Physical Sciences and Mathematics - Mathematics - Geometry and Topology
Principal Investigator:Ali Tahzibi
Grantee:Ali Tahzibi
Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
Pesquisadores principais:
Daniel Smania Brandão
Associated researchers:Carlos Alberto Maquera Apaza ; Douglas Coates ; Gabriel Ponce ; José Régis Azevedo Varão Filho
Associated grant(s):23/00676-6 - Entropy of discretized Anosov flows, AP.R SPRINT
19/09403-7 - The role of unstable entropy in smooth ergodic theory, AV.EXT
Associated scholarship(s):22/16259-2 - Statistical and non-statistical properties of dynamical systems, BP.PD
23/03954-7 - Empirical measures in Dynamics, BP.MS
22/16016-2 - Conservative Dynamical Systems, BP.IC
+ associated scholarships 22/15044-2 - Entropy in dynamical systems and information theory, BP.IC
22/05300-1 - Dissipative particles under the action of expanding dynamical systems, BP.DR
22/10589-0 - Introduction to dynamical systems through examples, BP.IC
22/10341-9 - Distribution of random orbits according to an IFS, BP.PD
18/18990-0 - Regularity of Lyapunov exponents and invariance principle, BP.PD
18/09974-1 - Conformal geometry and complex one-dimensional dynamical systems, BP.IC
18/04076-5 - Margulis measures and partially hyperbolic systems, BP.DR - associated scholarships

Abstract

In general terms, the objective of research in the area of Dynamical systems and Ergodic Theory is to describe the long term behavior of systems with an evolution law. There are deep interactions among Dynamical systems, other areas in mathematics like probability theory, geometry and number theory. There are also connections with many sub-areas in physics. In the Smooth ergodic theory, typically one tries to compare the evolution of an observable by action of a group with the long term behavior of independent random variables. In the last 60s and 70s many fundamental results in smooth ergodic theory were obtained and the vast majority were obtained in the context of uniformly hyperbolic dynamics which was mainly introduced by S. Smale (without forgetting Andronov and Pontryagin and Moscow school). Bowen, Ruelle and Sinai realized a connection between statistical mechanic and uniformly hyperbolic dynamics and constructed physical measures for such systems. After progresses in uniformly hyperbolic dynamics, weak forms of hyperbolicity like dominated splitting, partial hyperbolicity and non-uniform hyperbolicity were introduced mainly by Mañé, Pugh, Pesin and Shub. In the one-dimensional dynamical systems, inspirations from statistical mechanics (Feigenbaum, Coullet and Tresser) yields the origins of a rich teory of renormalization operator (defined dynamically) and universality in dynamics. Sullivan and Lyubich developped in a fundamental way the theory and it is interesting to mention that the Lyubich works, the (Smale) Horseshoe appears in the infinite dimensional model dynamics. The ideas from ergodic theory have been used in a very effective way to solve problems in number theory and Lie algebras too. We may highlight ergodic methods in the study of geodesic flow and horocyclic flow in the manifolds with negative curvature. The solution of Margulis for the oppenheim conjecture is also based on the methods from smooth ergodic theory and dynamical systems. This project is divided into following sub-projects which have both probabilistic and algebraic aspects. Possible solutions to the questions will have a positive impact in the development of the theory: Ergodic properties of dynamical systems preserving a "natural" probability measure: Equilibrium states for partially hyperbolic diffeomorphisms (in particular measures of maximal entropy, Margulis measures)Stable ergodicity for conservative diffeomorphisms Equilibrium states for action of groups Piecewise expanding transformations with low regularity Comparing classical partially hyperbolic dynamical systems and random walks, Invariance Principle (introduced by Avila-Viana generalizing Furstenberg, Ledrappier (and others) results into non-linear dynamical setting)Renormalization Theory and universality in one-dimensional dynamics Deformation theory in one-dimensional dynamics Dynamics defined by action of groups: Anosov (sub)systems and their representationsin SL(n,R) (AU)

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Scientific publications (14)
(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)
TAHZIBI, ALI. Unstable entropy in smooth ergodic theory {*}. Nonlinearity, v. 34, n. 8, p. R75-R118, . (17/06463-3)
CANTARINO, MARISA; VARAO, REGIS. Anosov endomorphisms on the two-torus: regularity of foliations and rigidity. Nonlinearity, v. 36, n. 10, p. 24-pg., . (18/13481-0, 17/06463-3)
EUZEBIO, RODRIGO D.; JUCA, JOABY S.; VARAO, REGIS. There Exist Transitive Piecewise Smooth Vector Fields on but Not Robustly Transitive. JOURNAL OF NONLINEAR SCIENCE, v. 32, n. 4, p. 15-pg., . (17/06463-3, 16/22475-9)
SMANIA, DANIEL. BESOV-ISH SPACES THROUGH ATOMIC DECOMPOSITION. ANALYSIS & PDE, v. 15, n. 1, p. 53-pg., . (17/06463-3)
SMANIA, DANIEL. Solenoidal attractors with bounded combinatorics are shy. ANNALS OF MATHEMATICS, v. 191, n. 1, p. 1-79, . (03/03107-9, 10/08654-1, 17/06463-3, 08/02841-4)
PONCE, G.; TAHZIBI, A.; VARAO, R.. On the Bernoulli property for certain partially hyperbolic diffeomorphisms. ADVANCES IN MATHEMATICS, v. 329, p. 329-360, . (09/16792-8, 15/02731-8, 11/21214-3, 16/05384-0, 14/23485-2, 12/14620-8, 17/06463-3, 16/22475-9)
ANTUNES, ANDRE AMARAL; CARVALHO, TIAGO; VARAO, REGIS. On topological entropy of piecewise smooth vector fields. Journal of Differential Equations, v. 362, p. 22-pg., . (19/10450-0, 16/22475-9, 19/10269-3, 21/12395-6, 17/18255-6, 17/06463-3)
NOVAES, DOUGLAS D.; VARAO, REGIS. A note on invariant measures for Filippov systems. BULLETIN DES SCIENCES MATHEMATIQUES, v. 167, . (18/13481-0, 17/06463-3, 16/22475-9)
ALVAREZ, CARLOS F.; SANCHEZ, ADRIANA; VARAO, REGIS. Equilibrium states for maps isotopic to Anosov. Nonlinearity, v. 34, n. 6, p. 4264-4282, . (18/18990-0, 16/22475-9, 17/06463-3)
BALADI, VIVIANE; SMANIA, DANIEL. Fractional Susceptibility Functions for the Quadratic Family: Misiurewicz-Thurston Parameters. Communications in Mathematical Physics, v. 385, n. 3, p. 1957-2007, . (17/06463-3)
CRISOSTOMO, J.; TAHZIBI, A.. Equilibrium states for partially hyperbolic diffeomorphisms with hyperbolic linear part. Nonlinearity, v. 32, n. 2, p. 584-602, . (14/23485-2, 17/06463-3, 15/15127-1, 13/03735-1)
DE LIMA, AMANDA; SMANIA, DANIEL. Central limit theorem for generalized Weierstrass functions. Stochastics and Dynamics, v. 19, n. 1, . (10/17419-6, 17/06463-3)
ESTEVEZ, GABRIELA; SMANIA, DANIEL; YAMPOLSKY, MICHAEL. Renormalization of Analytic Multicritical Circle Maps with Bounded Type Rotation Numbers. BULLETIN OF THE BRAZILIAN MATHEMATICAL SOCIETY, v. N/A, p. 19-pg., . (17/06463-3)
SMANIA, DANIEL. Classic and Exotic Besov Spaces Induced by Good Grids. JOURNAL OF GEOMETRIC ANALYSIS, v. 31, n. 3, . (17/06463-3)

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