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Infinite dimensional hyperbolic dynamics

Grant number: 13/20912-4
Support type:Regular Research Grants
Duration: January 01, 2014 - June 30, 2016
Field of knowledge:Physical Sciences and Mathematics - Mathematics
Principal Investigator:Joachim Weber
Grantee:Joachim Weber
Home Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil

Abstract

Consider the infinite dimensional hyperbolic dynamical system given by the heat (semi-)flow on the loop space of a closed Riemannian manifold (in the presence of a generic perturbation). The recently discovered *backward* lambda-Lemma is expected to be the key substitute for the non-existing backward flow. The lack of a backward flow caused the field to dry out in the mid 80s after those results that only use a forward flow had been carried over from the finite dimensional situation. One main application will be to construct global stable foliations, because these -- together with the known unstable foliations -- turned out to be the fundamental tools in the finite dimensional case. In finite dimensions stable and unstable foliations were constructed by Palis in 1969. Their significance lies in the fact that they provide natural coordinate systems near hyperbolic singularities. Another major application of the backward lambda-Lemma will be to calculate the Morse homology associated to a *semi*-flow. This will generalize the method of Abbondandolo-Majer which only applies to genuine flows. In fact, our purpose is to make the Morse homology tool available to the field of geometric analysis where many examples of semi-flows arise. (AU)

Scientific publications
(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)
WEBER, JOA. Stable foliations and semi-flow Morse homology. ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA-CLASSE DI SCIENZE, v. 17, n. 3, p. 853-909, 2017. Web of Science Citations: 1.
WEBER, JOA. CLASSICAL MORSE THEORY REVISITED - I BACKWARD lambda-LEMMA AND HOMOTOPY TYPE. TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS, v. 47, n. 2, p. 641-646, JUN 2016. Web of Science Citations: 2.
WEBER, JOA. Contraction method and Lambda-Lemma. SAO PAULO JOURNAL OF MATHEMATICAL SCIENCES, v. 9, n. 2, SI, p. 263-298, DEC 2015. Web of Science Citations: 2.

Please report errors in scientific publications list by writing to: cdi@fapesp.br.