When a completely integrable Hamiltonian system h, which is written in action-angle coordinates, ... more When a completely integrable Hamiltonian system h, which is written in action-angle coordinates, is perturbed, the action variables remain stable over exponentially long time intervals. The hypotheses are the quasi-convexity of h and the Gevrey- regularity of h and of the perturbation, with 1. This is a generalization of Nekhoroshev's Theorem (which corresponds to the analytic case, = 1). The stability time is governed by an exponent which can be chosen to be 1=2n in general, but which can be improved in 1=2(n 2) for the orbits passing close to a double resonance of h. For > 1, the existence of Gevrey- functions with compact support allows us to prove the optimality of the previous result: for three degrees of freedom or more, we construct systems exhibiting unstable orbits for which the speed of drift is optimal. These results were obtained in a joint work with Jean-Pierre Marco which started as a collaboration with Michael Herman (MS03a).
We consider the susceptibility function Psi(z) of a piecewise expanding unimodal interval map f w... more We consider the susceptibility function Psi(z) of a piecewise expanding unimodal interval map f with unique acim mu, a perturbation X, and an observable phi. Combining previous results (deduced from spectral properties of Ruelle transfer operators) with recent work of Breuer-Simon (based on techniques from the spectral theory of Jacobi matrices and a classical paper of Agmon), we show that density of the postcritical orbit (a generic condition) implies that Psi(z) has a strong natural boundary on the unit circle. The Breuer-Simon method provides uncountably many candidates for the outer functions of Psi(z), associated to precritical orbits. If the perturbation X is horizontal, a generic condition (Birkhoff typicality of the postcritical orbit) implies that the nontangential limit of the Psi(z) as z tends to 1 exists and coincides with the derivative of the acim with respect to the map (linear response formula). Applying the Wiener-Wintner theorem, we study the singularity type of nontangential limits as z tends to e^{i\omega}. An additional LIL typicality assumption on the postcritical orbit gives stronger results.
We consider the behaviour near resonances of linearizations of germs of holomorphic diffeomorphis... more We consider the behaviour near resonances of linearizations of germs of holomorphic diffeomorphisms of.C; 0/ and of the semi-standard map. We prove that for each resonance there exists a suitable blow-up of the Taylor series of the linearization under which it converges uniformly to an analytic function as the multiplier, or rotation number, tends non-tangentially to the resonance. This limit
The McMillan map is a one-parameter family of integrable sym- plectic maps of the plane, for whic... more The McMillan map is a one-parameter family of integrable sym- plectic maps of the plane, for which the origin is a hyperbolic fixed point with a homoclinic loop, with small Lyapunov exponent when the parameter is small. We consider a perturbation of the McMillan map for which we show that the loop breaks in two invariant curves which are exponentially
When a completely integrable Hamiltonian system h, which is written in action-angle coordinates, ... more When a completely integrable Hamiltonian system h, which is written in action-angle coordinates, is perturbed, the action variables remain stable over exponentially long time intervals. The hypotheses are the quasi-convexity of h and the Gevrey- regularity of h and of the perturbation, with 1. This is a generalization of Nekhoroshev's Theorem (which corresponds to the analytic case, = 1). The stability time is governed by an exponent which can be chosen to be 1=2n in general, but which can be improved in 1=2(n 2) for the orbits passing close to a double resonance of h. For > 1, the existence of Gevrey- functions with compact support allows us to prove the optimality of the previous result: for three degrees of freedom or more, we construct systems exhibiting unstable orbits for which the speed of drift is optimal. These results were obtained in a joint work with Jean-Pierre Marco which started as a collaboration with Michael Herman (MS03a).
We consider the susceptibility function Psi(z) of a piecewise expanding unimodal interval map f w... more We consider the susceptibility function Psi(z) of a piecewise expanding unimodal interval map f with unique acim mu, a perturbation X, and an observable phi. Combining previous results (deduced from spectral properties of Ruelle transfer operators) with recent work of Breuer-Simon (based on techniques from the spectral theory of Jacobi matrices and a classical paper of Agmon), we show that density of the postcritical orbit (a generic condition) implies that Psi(z) has a strong natural boundary on the unit circle. The Breuer-Simon method provides uncountably many candidates for the outer functions of Psi(z), associated to precritical orbits. If the perturbation X is horizontal, a generic condition (Birkhoff typicality of the postcritical orbit) implies that the nontangential limit of the Psi(z) as z tends to 1 exists and coincides with the derivative of the acim with respect to the map (linear response formula). Applying the Wiener-Wintner theorem, we study the singularity type of nontangential limits as z tends to e^{i\omega}. An additional LIL typicality assumption on the postcritical orbit gives stronger results.
We consider the behaviour near resonances of linearizations of germs of holomorphic diffeomorphis... more We consider the behaviour near resonances of linearizations of germs of holomorphic diffeomorphisms of.C; 0/ and of the semi-standard map. We prove that for each resonance there exists a suitable blow-up of the Taylor series of the linearization under which it converges uniformly to an analytic function as the multiplier, or rotation number, tends non-tangentially to the resonance. This limit
The McMillan map is a one-parameter family of integrable sym- plectic maps of the plane, for whic... more The McMillan map is a one-parameter family of integrable sym- plectic maps of the plane, for which the origin is a hyperbolic fixed point with a homoclinic loop, with small Lyapunov exponent when the parameter is small. We consider a perturbation of the McMillan map for which we show that the loop breaks in two invariant curves which are exponentially
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