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Journal of Dynamical and Control Systems

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01-10-2013

Generic Transitivity for Couples of Hamiltonians

Author: Vito Mandorino

Published in: Journal of Dynamical and Control Systems | Issue 4/2013

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Abstract

We study orbits and reachable sets of generic couples of Hamiltonians H ₁, H ₂ on a symplectic manifold N. We prove that, C ^k-generically for k large enough, orbits coincide with the whole of N and that the same is true for reachable sets when N is compact. Our results are stated in terms of a strong form of genericity which makes use of the notion of rectifiable subsets of positive codimension in Banach or Fréchet spaces.

previous article On the Stokes Phenomenon of a Family of Multi-Perturbed Level-One Meromorphic Linear Differential Systems

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This should be more properly called a pseudogroup, since the flows may not be complete.

It is also stronger than some other notions of translational invariant ‘smallness’ in infinite-dimensional spaces, such as prevalence or Aronszajn nullity (see [3]).

We will always assume that, given H ₁ as above, the family \(\{N^{l}\}_l\) has been chosen once for all. Such a choice will never play any role.

In [2] the terminology ‘Poisson stable’ rather than ‘non-wandering’ is used.

A Fredholm map of index i between separable Banach spaces is a C ¹ map such that the differential is Fredholm of index i at every point (recall that the index is locally constant).

Recall that in Assumption 1, we defined T N ₀ as the union \(\bigcup_{l\in{\mathbb{N}}}T N^{l}\) where \(\{N^{l}\}_l\) is a once-for-all fixed countable family of submanifolds of codimension greater than or equal to codim N ₀ and whose union covers N ₀.

Note that ψ also induces a change of coordinates on J ^k − 1(N,ℝ), which is of class C ¹ because ψ is of class C ^k. Hence, the codimension of W′ is the same as the codimension of its image under this diffeomorphism. This legitimates the subsequent computations (and accounts for the requirement \(H_1\in C^{\ k+1}\) rather than just C ^k).

Note that ψ×id also induces a local diffeomorphism of class C ¹ from (a subset of) J ^k − 1(N×ℝ,ℝ) to J ^k − 1(U×ℝ,ℝ), thus preserving codimensions of subsets.

For reasons of page layout, the matrix A(j ′) written here is rather the transpose of what it should be according to the matrix in Eq. 4.1.

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Title: Generic Transitivity for Couples of Hamiltonians
Author: Vito Mandorino
Publication date: 01-10-2013
Publisher: Springer US
Published in: Journal of Dynamical and Control Systems / Issue 4/2013
Print ISSN: 1079-2724
Electronic ISSN: 1573-8698
DOI: https://doi.org/10.1007/s10883-013-9197-0

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