Quasianalytic Monogenic Solutions of a Cohomological Equation

We prove that the solutions of a cohomological equation of complex dimension one and in the analytic category have a monogenic dependence on the parameter, and we investigate the question of their quasianalyticity. This cohomological equation is the standard linearized conjugacy equation for germs of holomorphic maps in a neighborhood of a fixed point. The parameter is the eigenvalue of the linear part, denoted by q. Borel\u2019s theory of non-analytic monogenic functions has been first investigated by Arnol\u2019d and Herman in the related context of the problem of linearization of analytic diffeomorphisms of the circle close to a rotation. Herman raised the question whether the solutions of the cohomological equation had a quasianalytic dependence on the parameter q. Indeed they are analytic for q 08 CS 1 , the unit circle S 1 appears as a natural boundary (because of resonances, i.e. roots of unity), but the solutions are still defined at points of S 1 which lie \u201cfar enough from resonances\u201d. We adapt to our case Herman\u2019s construction of an increasing sequence of compacts which avoid resonances and prove that the solutions of our equation belong to the associated space of monogenic functions ; some general properties of these monogenic functions and particular properties of the solutions are then studied. For instance the solutions are defined and admit asymptotic expansions at the points of S 1 which satisfy some arithmetical condition, and the classical Carleman Theorem allows us to answer negatively to the question of quasianalyticity at these points. But resonances (roots of unity) also lead to asymptotic expansions, for which quasianalyticity is obtained as a particular case of Ecalle\u2019s \ub4 theory of resurgent functions. And at constant-type points, where no quasianalytic Carleman class contains the solutions, one can still recover the solutions from their asymptotic expansions and obtain a special kind of quasianalyticity. Our results are obtained by reducing the problem, by means of Hadamard\u2019s product, to the study of a fundamental solution (which turns out to be the so-called q-logarithm or \u201cquantum logarithm\u201d). We deduce as a corollary of our work the proof of a conjecture of Gammel on the monogenic and quasianalytic properties of a certain number-theoretical Borel-Wolff-Denjoy series

Natural boundary for the susceptibility function of generic piecewise expanding unimodal maps

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.Comment: LaTex, 23 pages, to appear ETD

Resurgence of inner solutions for perturbations of the McMillan map

A sequence of “inner equations” attached to certain perturbations of the McMillan map was considered in [MSS09], their solutions were used in that article to measure an exponentially small separatrix splitting. We prove here all the results relative to these equations which are necessary to complete the proof of the main result of [MSS09]. The present work relies on ideas from resurgence theory: we describe the formal solutions, study the analyticity of their Borel transforms and use ´Ecalle’s alien derivations to measure the discrepancy between different Borel-Laplace sums.Preprin

Continuation of the exponentially small transversality for the splitting of separatrices to a whiskered torus with silver ratio

We study the exponentially small splitting of invariant manifolds of whiskered (hyperbolic) tori with two fast frequencies in nearly-integrable Hamiltonian systems whose hyperbolic part is given by a pendulum. We consider a torus whose frequency ratio is the silver number $\Omega=\sqrt{2}-1$. We show that the Poincar\'e-Melnikov method can be applied to establish the existence of 4 transverse homoclinic orbits to the whiskered torus, and provide asymptotic estimates for the tranversality of the splitting whose dependence on the perturbation parameter $\varepsilon$ satisfies a periodicity property. We also prove the continuation of the transversality of the homoclinic orbits for all the sufficiently small values of $\varepsilon$, generalizing the results previously known for the golden number.Comment: 17 pages, 2 figure

A new method for measuring the splitting of invariant manifolds

We study the so-called Generalized Arnol’d Model (a weakly hyperbolic near-integrable Hamiltonian system), with d+ 1 degrees of freedom (d 2), in the case where the perturbative term does not affect a fixed invariant d-dimensional torus. This torus is thus independent of the two perturbation parameters which are denoted ε (ε> 0) and µ. We describe its stable and unstable manifolds by solutions of the Hamilton–Jacobi equation for which we obtain a large enough domain of analyticity. The splitting of the manifolds is measured by the partial derivatives of the difference ∆S of the solutions, for which we obtain upper bounds which are exponentially small with respect to ε. A crucial tool of the method is a characteristic vector field, which is defined on a part of the configuration space, which acts by zero on the function ∆S and which has constant coefficients in well-chosen coordinates. It is in the case where |µ| is bounded by some positive power of ε that the most precise results are obtained. In a particular case with three degrees of freedom, the method leads also to lower bounds for the splitting

Quasianalytic monogenic solutions of a cohomological equation

We prove that the solutions of a cohomological equation of complex dimension one and in the analytic category have a monogenic dependence on the parameter, and we investigate the question of their quasianalyticity. This cohomological equation is the standard linearized conjugacy equation for germs of holomorphic maps in a neighborhood of a fixed point. The parameter is the eigenvalue of the linear part, denoted by q. Borel’s theory of non-analytic monogenic functions has been first investigated by Arnol’d and Herman in the related context of the problem of linearization of analytic diffeomorphisms of the circle close to a rotation. Herman raised the question whether the solutions of the cohomological equation had a quasianalytic dependence on the parameter q. Indeed they are analytic for q ∈ CS 1 , the unit circle S 1 appears as a natural boundary (because of resonances, i.e. roots of unity), but the solutions are still defined at points of S 1 which lie “far enough from resonances”. We adapt to our case Herman’s construction of an increasing sequence of compacts which avoid resonances and prove that the solutions of our equation belong to the associated space of monogenic functions ; some general properties of these monogenic functions and particular properties of the solutions are then studied. For instance the solutions are defined and admit asymptotic expansions at the points of S 1 which satisfy some arithmetical condition, and the classical Carleman Theorem allows us to answer negatively to the question of quasianalyticity at these points. But resonances (roots of unity) also lead to asymptotic expansions, for which quasianalyticity is obtained as a particular case of Ecalle’s ´ theory of resurgent functions. And at constant-type points, where no quasianalytic Carleman class contains the solutions, one can still recover the solutions from their asymptotic expansions and obtain a special kind of quasianalyticity. Our results are obtained by reducing the problem, by means of Hadamard’s product, to the study of a fundamental solution (which turns out to be the so-called q-logarithm or “quantum logarithm”). We deduce as a corollary of our work the proof of a conjecture of Gammel on the monogenic and quasianalytic properties of a certain number-theoretical Borel-Wolff-Denjoy series

Limit at resonances of linearizations of some complex analytic dynamical systems

We consider the behaviour near resonances of linearizations of germs of holomorphic diffeomorphisms of $({\Bbb C},0)$ 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 function is explicitly computed and related to questions of formal classification, both for the case of germs and for the case of the semi-standard map

Natural boundary for the susceptibility function of generic piecewise expanding unimodal maps

LaTex, 23 pages, ETDS, 34 (2014) 777-800International audienceWe 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