Dynamic in low dimensions

• Grant number: 06/03829-2
• Support Opportunities: Research Projects - Thematic Grants
• Start date: December 01, 2006
• End date: November 30, 2011
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: André Salles de Carvalho
• Grantee: André Salles de Carvalho
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Pesquisadores principais: Albert Meads Fisher ; Edson de Faria ; Edson Vargas ; Salvador Addas Zanata
• Associated researchers: Eduardo Colli ; Fábio Armando Tal ; Pedro Antonio Santoro Salomão ; Salvador Addas Zanata
• Associated research grant(s): 11/01830-1 - Dynamics and contact topology, AV.EXT; 11/50128-8 - Charles Tresser | IBM - Estados Unidos, AV.EXT; 10/09667-0 - Topological methods in low-dimensional dynamics, AV.EXT; 09/17358-0 - The transition from finite to infinite measures in dynamical systems, AV.EXT
• Associated scholarship(s): 11/19948-9 - Existence o finite energy foliations on symplectizations of contact manifolds, BE.PQ; 11/12133-0 - Infinite measures, self-similar actions and continued fractions in ergodic theory, BE.PQ; 10/20159-6 - Pruning theory and dynamics of the complex Hénon family, BP.PD; + associated scholarships 09/16234-5 - Geometric Cone-Structures on Manifolds of dimension 2 and 3, BP.PD; 09/08367-5 - Physical measures for critical covering maps of the circle, BP.DR; 08/10659-1 - Renormalisation in the Hénon family, BP.PD; 08/10059-4 - Ergodic Optimization, BP.PD; 07/08114-4 - Dynamics of Fuchsians groups, BP.IC - associated scholarships

Abstract

The modern theory of dynamical systems started with the work of Poincaré and, since then, grew into a mature and very active branch of mathematical research. The main goal of this project is to further the study of the following areas of dynamical systems theory: - Hamiltonian systems with two degrees of freedom, their dynamical and topological aspects. - Polinomial differential equations on the plane and the 16th Problem of Hilbert. - Two¬dimensional homeomorphisms and diffeomorphisms such as Hénon maps and twits maps of the annulus. Renormalization theory in dimensions 1 and 2. - Interval endomorphisms (e.g., delicate analytic questions such as decay of geometry and existence of invariant measures); critical circle mappings; renormalization and parameter space. - Teichmueller theory and connections with low dimensional dynamics. - Differentiable ergodic theory. While dynamical systems theory developed it also moved away from other branchs of mathematics which were also started by Poincaré: symplectic geometry and topology. Another important goal of this project is to look for and to develop connections between these areas to the point of making it possible to use techniques of each area to attack problems of the other. (AU)

Scientific publications

(12)

(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)

DE CARVALHO, ANDRE; HALL, TOBY. Riemann surfaces out of paper. PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY, v. 108, n. 3, p. 541-574, MAR 2014. (06/03829-2)

BOYLAND, PHILIP; DE CARVALHO, ANDRE; HALL, TOBY. Inverse limits as attractors in parameterized families. BULLETIN OF THE LONDON MATHEMATICAL SOCIETY, v. 45, n. 5, p. 1075-1085, OCT 2013. (06/03829-2, 10/09667-0)

DE FARIA, EDSON; TRESSER, CHARLES. Bell Inequality Violations Under Reasonable and Under Weak Hypotheses. Physical Review Letters, v. 110, n. 26, JUN 28 2013. (06/03829-2, 11/50128-8)

DE CARVALHO, ANDRE; HALL, TOBY. DECORATION INVARIANTS FOR HORSESHOE BRAIDS. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS, v. 27, n. 3, p. 863-906, JUL 2010. (06/03829-2)

HRYNIEWICZ, UMBERTO; SALOMAO, PEDRO A. S.. ON THE EXISTENCE OF DISK-LIKE GLOBAL SECTIONS FOR REEB FLOWS ON THE TIGHT 3-SPHERE. Duke Mathematical Journal, v. 160, n. 3, p. 415-465, 2011. (06/03829-2)

DE CARVALHO, ANDRE; HALL, TOBY. Riemann surfaces out of paper. PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY, v. 108, p. 34-pg., 2014-03-01. (06/03829-2)

BOYLAND, PHILIP; DE CARVALHO, ANDRE; HALL, TOBY. Inverse limits as attractors in parameterized families. BULLETIN OF THE LONDON MATHEMATICAL SOCIETY, v. 45, p. 11-pg., 2013-10-01. (10/09667-0, 06/03829-2)

DE MELO, WELINGTON; SALOMAO, PEDRO A. S.; VARGAS, EDSON. A full family of multimodal maps on the circle. Ergodic Theory and Dynamical Systems, v. 31, n. 5, p. 1325-1344, OCT 2011. (06/03829-2)

DE CARVALHO, ANDRE; HALL, TOBY. Paper surfaces and dynamical limits. PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA, v. 107, n. 32, p. 14030-14035, AUG 10 2010. (06/03829-2)

DE CARVALHO, ANDRE; HALL, TOBY. Paper folding, Riemann surfaces and convergence of pseudo-Anosov sequences. GEOMETRY & TOPOLOGY, v. 16, n. 4, p. 1881-1966, 2012. (06/03829-2)

DE CARVALHO, ANDRE; HALL, TOBY; VENZKE, RUPERT. ON PERIOD MINIMAL PSEUDO-ANOSOV BRAIDS. Proceedings of the American Mathematical Society, v. 137, n. 5, p. 1771-1776, 2009. (06/03829-2)

DE MELO, WELINGTON; SALOMAO, PEDRO A. S.; VARGAS, EDSON. A full family of multimodal maps on the circle. Ergodic Theory and Dynamical Systems, v. 31, p. 20-pg., 2011-10-01. (06/03829-2)