On the regularity of Mather's β-function for standard-like twist maps
Introduction
In this note we would like to investigate some regularity properties of the so-called Mather's β-function (or minimal average action) for twist maps of the annulus. This object is related to the minimal average action of configurations with a prescribed rotation number (the so-called Aubry-Mather orbits) and plays a crucial role in the study of the dynamics of twist maps; see section 2 for a more detailed introduction. In particular, many intriguing questions and conjectures related to problems in dynamics, analysis and geometry have been translated into questions about this function and its regularity properties (see for example [15], [22], [23], [25], [26] and references therein), shedding a new light on these issues and, in some cases, paving the way for their solution.
Two of the main questions that underpin our current interest in the subject are the following:
a) Do regularity properties of β-function (i.e., differentiability, higher smoothness, etc.) allow one to infer any information on the dynamics of the system?
b) To which extent does this function identify the system? Does it satisfy any sort of rigidity property?
Despite the huge amount of attention that these questions have attracted over the past years—in particular, understanding its regularity and its implications—they remain essentially open. In the twist map case, the best result known is that this map is strictly convex and differentiable at all irrationals. Moreover, differentiability at a rational number is a very atypical phenomenon: it corresponds to the existence of an invariant circle consisting of periodic orbits whose rotation number is (see [18]). An extension of these results to surfaces was provided in [15].
Goal of this article is to address the regularity and uniqueness issues raised in a) and b), and provide some new interesting answers in the special case of standard-like maps. More specifically, our starting point is the paper [5] which establishes some rigidity properties of the complex extension of analytic parametrizations of KAM curves. We use the main result of [5] to build up a -holomorphic complex function which coincides with Mather's β function on the set of real diophantine frequencies, we prove that this extension is unique and deduce uniqueness results for Mather's β function. See Theorem 3.3 and Corollary 3.4 for precise statements. To the best of our knowledge, this complex extension of Mather's β function (that turns out to be canonical) has never been studied before and we believe that could be an important object for further investigation of the dynamics.
The article is organized as follows. In section 2 we provide a brief introduction to Aubry-Mather theory and introduce the main object of investigation (Definition 2.5). In section 3 we state our main results (Theorem 3.3 and Corollary 3.4), whose proofs will be detailed in section 5. Some auxiliary results will be described in section 4 and appendix A.
Acknowledgements. The authors acknowledge the support of the Centro di Ricerca Matematica Ennio de Giorgi and of UniCredit Bank R&D group for financial support through the Dynamics and Information Theory Institute at the Scuola Normale Superiore. CC, SM and AS acknowledge the support of MIUR PRIN Project Regular and stochastic behaviour in dynamical systems nr. 2017S35EHN. AS acknowledges the support of the MIUR Department of Excellence grant CUP E83C18000100006. DS thanks Fibonacci Laboratory for their hospitality. CC and AS have been partially supported by the GNAMPA group of the “Istituto Nazionale di Alta Matematica” (INdAM).
Section snippets
A synopsis of Aubry–Mather theory for twist maps of the cylinder
At the beginning of 1980's Serge Aubry and John Mather developed, independently, a novel and fruitful approach to the study of monotone twist maps of the annulus, based on the so-called principle of least action, nowadays commonly called Aubry–Mather theory. They pointed out the existence of global action-minimizing orbits for any given rotation number; these orbits minimize the discrete Lagrangian action with fixed end-points on all time intervals (for a more detailed introduction, see for
Statement of the main result
Let us now consider the framework of a standard-like twist map (see Example 2 in Section 2): with g a 1-periodic, real analytic function of zero mean. Let G be the primitive of g with zero mean, and observe that G is real analytic and 1-periodic as well. As a generating function for , we take As was mentioned earlier, Mather's β-function at any is defined as the average action of any minimal configuration of rotation
Intermediate results
In order to prove Theorem 3.3, let us first recall part of the results of [5].
A parametrized invariant curve of rotation number ω for is a pair of continuous functions such that Note that, if is a parametrized invariant curve for of rotation number , then is a minimal configuration of rotation number ω and the limit in equation (7) becomes
Proof of Theorem 3.3
We now give ourselves and define , with and as in Corollary 4.3. We suppose that g and M satisfy the assumptions of Theorem 3.3 with this value of c. We must find a function satisfying all the claims of Theorem 3.3.
Among our assumptions, we have , therefore and , whence . We can thus apply Corollary 4.3 and use the function satisfying equation (22) as well as the properties
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