Journal of Modern Dynamics (JMD)

Connecting orbits for families of Tonelli Hamiltonians

Pages: 499 - 538, Issue 4, October 2012      doi:10.3934/jmd.2012.6.499

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Vito Mandorino - Université Paris-Dauphine, CEREMADE UMR CNRS 7534, Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France (email)

Abstract: We investigate the existence of Arnold diffusion-type orbits for systems obtained by iterating in any order the time-one maps of a family of Tonelli Hamiltonians. Such systems are known as 'polysystems' or 'iterated function systems'. When specialized to families of twist maps on the cylinder, our results are similar to those obtained by Moeckel [20] and Le Calvez [15]. Our approach is based on weak KAM theory and is close to the one used by Bernard in [3] to study the case of a single Tonelli Hamiltonian.

Keywords:  Arnold diffusion, Tonelli Hamiltonian, heteroclinic orbit, iterated function system, weak KAM theory, Lax--Oleinik operator.
Mathematics Subject Classification:  Primary: 37J50, 37J45; Secondary: 37J40, 70Hxx.

Received: May 2012;      Available Online: January 2013.