We consider generic properties of Lagrangians. Our main result is the Theorem of Kupka-Smale, in the Lagrangian setting, claiming that, for a convex and superlinear Lagrangian defined in a compact surface, for each k ∈ R, generically, in Mañé's sense, the energy level, k, is regular and all periodic orbits, in this level, are nondegenerate at all orders, that is, the linearized Poincaré map, restricted to this energy level, does not have roots of the unity as eigenvalues. Moreover, all heteroclinic intersections in this level are transversal. All the results that we present here are true in dimension n ≥ 2, except one (Theorem 18), whose proof we are able to obtain just for dimension 2. * Supported by CAPES, scholarship.Lagrangian submanifolds in order to get the transversality of heteroclinic intersections.The Kupka-Smale Theorem , in this formulation, resembles the Bumpy Metrics Theorem, for geodesic flows, formulated by R. Abraham in 1968, and proved by D. V. Anosov in 1983 (Anosov, [4]).The work of W. Klingenberg and F. Takens [12] in the Bumpy Metrics Theorem proof was corrected by Anosov [4] using an induction method similar to the one used by M. Peixoto [16] in the proof of the classical Kupka-Smale Theorem.In this work, we will employ the same techniques used by Anosov [4], adapted to perturbations by potentials. In order to apply to the case of perturbation by potentials, it is necessary to introduce a modification of the standard argument in the Control Theory for differential equations, initially used by Klingenberg [11], for geodesic flow perturbation setting, by J. A. Miranda [14] for magnetic flows on surfaces and by Contreras [7] for the proof of Franks' Lemma for geodesic flows. In this case we do not have especial coordinates, like Fermi coordinates, as in [11], [14] and [7], thus we introduce a new method without the use of tubular neighborhoods, that solves this trouble. In the beginning of the proof we use an argument similar to the one used by Robinson [17], Lemma 19, Pg. 592, but the proof is quite different.Observe that the generic properties in Mañé's sense cannot be obtained from the pioneering work of Robinson in the general Hamiltonian setting (see [17] and [18]) because the set of all Hamiltonians is bigger than the set of all potentials in M .The transversality is the easy part. Here we follow the approach of Contreras & Paternain [9], Lemma 2.6 or J.A. Miranda [14], Lemma 3.9. The problem in this case is to construct an explicit potential that represents the perturbation.The main obstacle in the proof of the Kupka-Smale Theorem in dimension n > 2 is the nature of the perturbation constructed. As in Contreras [7], Lemma 7.3 and 7.4, we need to solve a matrix equation in the Lie algebra of the symplectic group S p (n) = {Symplectic matrices 2n × 2n}. The solubility of this equation is strongly related with the existence of repeated eigenvalues of the matrix H pp in local coordinates. The problem is that we can not change this characteristic by adding a potential. Moreover, the equati...

Consider the shift σ acting on the Bernoulli space Σ = {1, 2, ..., n} N . We denoteΣ = {1, 2, ..., n} Z = Σ × Σ. We analyze several properties of the maximizing probability µ ∞

Abstract. Consider the shift T acting on the Bernoulli space Σ = {1, 2, 3, .., d} N and A : Σ → R a Holder potential. Denote m(A) = max ν an invariant probability for T

We consider a piecewise analytic real expanding map f : [0, 1] → [0, 1] of degree d which preserves orientation, and a real analytic positive potential g : [0, 1] → R. We assume the map and the potential have a complex analytic extension to a neighborhood of the interval in the complex plane. We also assume log g is well defined for this extension.It is known in Complex Dynamics that under the above hypothesis, for the given potential β log g, where β is a real constant, there exists a real analytic eigenfunction φ β defined on [0, 1] (with a complex analytic extension) for the Ruelle operator of β log g.Under some assumptions we show that 1 β log φ β converges and is a piecewise analytic calibrated subaction.Our theory can be applied when log g(x) = − log f ′ (x). In that case we relate the involution kernel to the so called scaling function.

In this work we introduce an idempotent pressure to level-2 functions and its associated density entropy. All this is related to idempotent pressure functions which is the natural concept that corresponds to the meaning of probability in the level-2 max-plus context. In this general framework the equilibrium states, maximizing the variational principle, are not unique. We investigate the connections with the general convex pressure introduced recently to level-1 functions by Biś, Carvalho, Mendes and Varandas. Our general setting contemplates the dynamical and not dynamical framework. We also study a characterization of the density entropy in order to get an idempotent pressure invariant by dynamical systems acting on probabilities; this is therefore a level-2 result. We are able to produce idempotent pressure functions at level-2 which are invariant by the dynamics of the pushforward map via a form of Ruelle operator.