Most Invariant Manifolds of Conservative Systems have Transitive Closure
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Most Invariant Manifolds of Conservative Systems have Transitive Closure Fábio Castro1 · Fernando Oliveira2 Accepted: 2 September 2020 © Foundation for Scientific Research and Technological Innovation 2020
Abstract The main result of this work in the following: for generic volume preserving flows on compact manifolds with the Cr topology, 1 ≤ r ≤ ∞ , the closure of every invariant manifold of periodic orbits and singularities is a chain transitive set. This was already known for generic symplectic and volume preserving diffeomorphisms. Keywords Chain transitivity · Invariant manifold · Genericity · Volume preserving flows
Introduction In differentiable dynamics, one of the most important objects after periodic orbits are the invariant manifolds. Homoclinic orbits also play an essential role. Homoclinic orbits are those obtained by the intersection of invariant manifolds of hyperbolic periodic orbits. Poincaré was probably the first one to draw the trellis of invariant manifolds that arises in the presence of a transverse homoclinic orbit, and became astonished with the complexity of the phenomenon. He conjectured that periodic orbits would be dense for most conservative systems. In his great Prize Memoir in the Acta Mathematica he dealt with the n body problem. He employed the method of analytic continuation and after some work found it to be insufficient. Then he turned his attention to the three body problem and wrote his seminal work Les Méthodes Nouvelles de la Mécanique Céleste [11]. By the end of his life he gave out his Last Geometric Theorem without a complete proof. It is very interesting to see how Birkhoff [4] describes in detail the history, events, achievements and failures in the work of Poincaré.
* Fábio Castro [email protected] Fernando Oliveira [email protected] 1
Departamento de Matemática Centro de Ciências Exatas, Universidade Federal do Espírito Santo, Vitória, Brazil
2
Departamento de Matemática, Instituto de Ciências Exatas, Universidade Federal de Minas Gerais, Belo Horizonte, Brazil
13
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Differential Equations and Dynamical Systems
The density of periodic orbits in the manifold is equivalent to the density of homoclinic orbits in invariant manifolds of hyperbolic periodic orbits. The general feeling is that this happens for most conservative systems. If this is the case, then the closure of each invariant manifold has a dense orbit. This is not known as well, even for surfaces. If we try something a little weaker like chain transitivity, then it is possible to prove that for symplectic and volume preserving diffeomorphisms of compact manifolds, generically the closure of invariant manifolds of hyperbolic periodic points are chain transitive sets [9]. All these problems make sense for conservative flows (volume preserving, hamiltonian, etc). The main result of this paper is the following: Main Theorem. Let M be a compact manifold without boundary and dimension greater or equal to 3. Let 𝜔 be a volume form on M and consider the set Xr𝜔 (M) of volume pr
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