Invariant curves for area preserving maps

This work aims to present tool of this approach is the so-called parameterization method, that produces a Newton-like iterative method to solve the invariance equation for an invariant torus, and the KAM theorem is a result on the convergence of the method. The method was introduced in this setting by A. González, À. Jorba, R. de la Llave and J. Villanueva [4] (see A tutorial on KAM theory for preliminary a version). In this work we have followed the review of the method exposed in A parameterizarion method for invariant manifolds

Factorization invariants in numerical monoids

Nonunique factorization in commutative monoids is often studied using factorization invariants, which assign to each monoid element a quantity determined by the factorization structure. For numerical monoids (co-finite, additive submonoids of the natural numbers), several factorization invariants have received much attention in the recent literature. In this survey article, we give an overview of the length set, elasticity, delta set, $\omega$-primality, and catenary degree invariants in the setting of numerical monoids. For each invariant, we present current major results in the literature and identify the primary open questions that remain

The First-Order Theory of Sets with Cardinality Constraints is Decidable

We show that the decidability of the first-order theory of the language that combines Boolean algebras of sets of uninterpreted elements with Presburger arithmetic operations. We thereby disprove a recent conjecture that this theory is undecidable. Our language allows relating the cardinalities of sets to the values of integer variables, and can distinguish finite and infinite sets. We use quantifier elimination to show the decidability and obtain an elementary upper bound on the complexity. Precise program analyses can use our decidability result to verify representation invariants of data structures that use an integer field to represent the number of stored elements.Comment: 18 page

Tonelli Hamiltonians without conjugate points and C0C^0 integrability

We prove that all the Tonelli Hamiltonians defined on the cotangent bundle T^*\T^n of the $n$-dimensional torus that have no conjugate points are $C^0$ integrable, i.e. T^*\T^n is $C^0$ foliated by a family \Fc of invariant $C^0$ Lagrangian graphs. Assuming that the Hamiltonian is $C^\infty$, we prove that there exists a $G_\delta$ subset \Gc of \Fc such that the dynamics restricted to every element of \Gc is strictly ergodic. Moreover, we prove that the Lyapunov exponents of every $C^0$ integrable Tonelli Hamiltonian are zero and deduce that the metric and topological entropies vanish.Comment: 37 page