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Ricardo Perez-Marco

Publications and source records attributed to Ricardo Perez-Marco.

27 records · Page 2Linked to original sources
arXiv2010

Hodge theory for Riemannian solenoids

A measured solenoid is a compact laminated space endowed with a transversal measure. The De Rham $L^2$-cohomology of the solenoid is defined by using differential forms which are smooth in the leafwise directions and $L^2$ in the transversal direction. We develop the theory of harmonic forms for Riemannian measured solenoids, and prove that this computes the De Rham $L^2$-cohomology of the solenoid. This implies in particular a Poincare duality result.

math.DG↗
arXiv2009

Ergodic solenoidal homology: density of ergodic solenoids

A measured solenoid is a laminated space endowed with a tranversal measure invariant by holonomy, as defined in arXiv:0910.2836. A measured solenoid immersed in a smooth manifold produces a closed current (known as generalized Ruelle-Sullivan current). Uniquely ergodic solenoids are those for which there is a unique (up to scalars) transversal measure. By the results in arXiv:0910.2913, for any smooth manifold, any real homology class is represented by a uniquely ergodic solenoid. In this paper, we prove that the currents associated to uniquely ergodic solenoids are dense in the space of closed currents, therefore proving the abundance of such objects.

math.DG↗
arXiv2009

Schwartzman cycles and ergodic solenoids

We extend Schwartzman theory beyond dimension 1 and provide a unified treatment of Ruelle-Sullivan and Schwartzman theories via Birkhoff's ergodic theorem for the class of immersions of solenoids with a trapping region.

math.DG↗
arXiv2009

Ergodic solenoidal homology: Realization theorem

We define generalized currents associated with immersions of abstract oriented solenoids with a transversal measure. We realize geometrically the full real homology of a compact manifold with these generalized currents, and more precisely with immersions of minimal uniquely ergodic solenoids. This makes precise and geometric De Rham's realization of the real homology by only using a restricted geometric subclass of currents.

math.DG↗
arXiv2000

Convergence or generic divergence of Birkhoff normal form

We prove that Birkhoff normal form of hamiltonian flows at a non-resonant singular point with given quadratic part are always convergent or generically divergent. The same result is proved for the normalization mapping and any formal first integral.

math.DS↗
arXiv2000

Linearization of holomorphic germs with resonant linear part

We study the linearization problem of germs of holomorphic diffeomorphisms with resonant linear part. The formal linearization requires in general an infinite number of algebraic relations to be satisfied by the coefficients of the power series defining the holomorphic germ. For a fixed germ, the linearizing mappings are not unique, but the existence of a finite jet with a formal divergent linearization but no convergent one forces all the other linearizing mappings to be diverging. We consider also polynomial families formally conjugated to a fixed resonant linear part. Then all the elements of the family are holomorphically linearizable, or we are in the precedent diverging situation for all but an exceptional set of parameter values of zero $Γ$-capacity. In the case of maximal non-trivial resonance we prove the optimality of Bruno condition.

math.DS↗
arXiv2000

A note on holomorphic extensions

We give a criterium of holomorphy for some type formal power series. This gives a stronger form of a Rothstein's type extension theorem for a particular ring of holomorphic functions.

math.DS↗
arXiv2000

Total convergence or general divergence in Small Divisors

We study generic holomorphic families of dynamical systems presenting problems of small divisors with fixed arithmetic. We prove that we have convergence for all parameter values or divergence everywhere except for an exceptional set in the potential theory sense. We illustrate this general principle in different problems of small divisors.

math.DS↗