Reports de recercahttp://hdl.handle.net/2117/322920191209T21:41:01Z20191209T21:41:01ZExponentially small splitting of separatrices associated to 3D whiskered tori with cubic frequenciesDelshams Valdés, AmadeuGonchenko, MarinaGutiérrez Serrés, Perehttp://hdl.handle.net/2117/13537520190627T05:12:45Z20190626T08:42:03ZExponentially small splitting of separatrices associated to 3D whiskered tori with cubic frequencies
Delshams Valdés, Amadeu; Gonchenko, Marina; Gutiérrez Serrés, Pere
We study the splitting of invariant manifolds of whiskered (hyperbolic) tori with three frequencies in a nearlyintegrable Hamiltonian system, whose hyperbolic part is given by a pendulum. We consider a 3dimensional toruswith a fast frequency vector¿/ve, with¿= (1,¿, ~¿) where ¿ is a cubic irrational number whose two conjugatesare complex, and the components of¿generate the fieldQ(¿). A paradigmatic case is the cubic golden vector,given by the (real) number ¿ satisfying ¿3= 1¿, and ~¿ = ¿2. For such 3dimensional frequency vectors,the standard theory of continued fractions cannot be applied, so we develop a methodology for determining thebehavior of the small divisors<k, ¿>,k¿Z3. Applying the PoincaréMelnikov method, this allows us tocarry outa careful study of the dominant harmonic (which depends one) of the Melnikov function, obtaining an asymptoticestimate for the maximal splitting distance, which is exponentially small ine, and valid for all sufficiently smallvalues ofe. This estimate behaves like exp{h1(e)/e1/6}and we provide, for the first time in a system with 3frequencies, an accurate description of the (positive) functionh1(e) in the numerator of the exponent, showing thatit can be explicitly constructed from the resonance properties of the frequency vector¿, and proving that it is aquasiperiodic function (and not periodic) with respect to lne. In this way, we emphasize the strong dependence ofthe estimates for the splitting on the arithmetic properties of the frequencies
20190626T08:42:03ZDelshams Valdés, AmadeuGonchenko, MarinaGutiérrez Serrés, PereWe study the splitting of invariant manifolds of whiskered (hyperbolic) tori with three frequencies in a nearlyintegrable Hamiltonian system, whose hyperbolic part is given by a pendulum. We consider a 3dimensional toruswith a fast frequency vector¿/ve, with¿= (1,¿, ~¿) where ¿ is a cubic irrational number whose two conjugatesare complex, and the components of¿generate the fieldQ(¿). A paradigmatic case is the cubic golden vector,given by the (real) number ¿ satisfying ¿3= 1¿, and ~¿ = ¿2. For such 3dimensional frequency vectors,the standard theory of continued fractions cannot be applied, so we develop a methodology for determining thebehavior of the small divisors<k, ¿>,k¿Z3. Applying the PoincaréMelnikov method, this allows us tocarry outa careful study of the dominant harmonic (which depends one) of the Melnikov function, obtaining an asymptoticestimate for the maximal splitting distance, which is exponentially small ine, and valid for all sufficiently smallvalues ofe. This estimate behaves like exp{h1(e)/e1/6}and we provide, for the first time in a system with 3frequencies, an accurate description of the (positive) functionh1(e) in the numerator of the exponent, showing thatit can be explicitly constructed from the resonance properties of the frequency vector¿, and proving that it is aquasiperiodic function (and not periodic) with respect to lne. In this way, we emphasize the strong dependence ofthe estimates for the splitting on the arithmetic properties of the frequenciesA note on symplectic and Poisson linearization of semisimple Lie algebra actionsMiranda Galcerán, Evahttp://hdl.handle.net/2117/2696020190124T11:06:53Z20150323T16:46:48ZA note on symplectic and Poisson linearization of semisimple Lie algebra actions
Miranda Galcerán, Eva
In this note we prove that an analytic symplectic action of a semisimple Lie algebra can be locally linearized in Darboux coordinates. This result yields simultaneous analytic linearization for Hamiltonian vector fields in a neighbourhood of a common zero. We also provide an example of smooth nonlinearizable Hamiltonian action with semisimple linear part. The smooth analogue only holds if the semisimple Lie algebra is of compact type. An analytic equivariant bDarboux theorem for bPoisson manifolds and an analytic equivariant Weinstein splitting theorem for general Poisson manifolds are also obtained in the Poisson setting
20150323T16:46:48ZMiranda Galcerán, EvaIn this note we prove that an analytic symplectic action of a semisimple Lie algebra can be locally linearized in Darboux coordinates. This result yields simultaneous analytic linearization for Hamiltonian vector fields in a neighbourhood of a common zero. We also provide an example of smooth nonlinearizable Hamiltonian action with semisimple linear part. The smooth analogue only holds if the semisimple Lie algebra is of compact type. An analytic equivariant bDarboux theorem for bPoisson manifolds and an analytic equivariant Weinstein splitting theorem for general Poisson manifolds are also obtained in the Poisson settingActionangle variables and a KAM theorem for bPoisson manifoldsKiesenhofer, AnnaMiranda Galcerán, EvaScott, Geoffreyhttp://hdl.handle.net/2117/2639020190124T11:06:29Z20150217T12:12:30ZActionangle variables and a KAM theorem for bPoisson manifolds
Kiesenhofer, Anna; Miranda Galcerán, Eva; Scott, Geoffrey
In this article we prove an actionangle theorem for bintegrable systems on bPoisson manifolds improving the actionangle theorem contained in [LMV11] for general Poisson manifolds in this setting. As an application, we prove a KAMtype theorem for bPoisson manifolds.
20150217T12:12:30ZKiesenhofer, AnnaMiranda Galcerán, EvaScott, GeoffreyIn this article we prove an actionangle theorem for bintegrable systems on bPoisson manifolds improving the actionangle theorem contained in [LMV11] for general Poisson manifolds in this setting. As an application, we prove a KAMtype theorem for bPoisson manifolds.Xiao's conjuecture for general fibred surfacesBarja Yáñez, Miguel ÁngelGonzález Alonso, VíctorNaranjo del Val, Joan Carleshttp://hdl.handle.net/2117/2499920190124T11:05:58Z20141211T12:05:29ZXiao's conjuecture for general fibred surfaces
Barja Yáñez, Miguel Ángel; González Alonso, Víctor; Naranjo del Val, Joan Carles
We prove that the genus g, the relative irregularity q_f and the Clifford index c_f of a nonisotrivial fibration
f satisfy the inequality q_f=gc_f. This gives in particular a proof of Xiao’s conjecture for
fibrations whose general fibres have maximal Clifford index.
Prerpint
20141211T12:05:29ZBarja Yáñez, Miguel ÁngelGonzález Alonso, VíctorNaranjo del Val, Joan CarlesWe prove that the genus g, the relative irregularity q_f and the Clifford index c_f of a nonisotrivial fibration
f satisfy the inequality q_f=gc_f. This gives in particular a proof of Xiao’s conjecture for
fibrations whose general fibres have maximal Clifford index.Stability and singularities of relative hypersurfacesBarja Yáñez, Miguel ÁngelStoppino, Lidiahttp://hdl.handle.net/2117/2499820190124T11:06:00Z20141211T12:01:17ZStability and singularities of relative hypersurfaces
Barja Yáñez, Miguel Ángel; Stoppino, Lidia
We study relative hypersurfaces, and prove an instability condition for the fibres. This
is the starting point for an investigation of the geometry of effective divisors on relative
projective bundles.
20141211T12:01:17ZBarja Yáñez, Miguel ÁngelStoppino, LidiaWe study relative hypersurfaces, and prove an instability condition for the fibres. This
is the starting point for an investigation of the geometry of effective divisors on relative
projective bundles.Algebraic description of Jacobians isogeneous to certain Prym varieties with polarization (1,2)Fedorov, YuriEnolski, Viktor Z.http://hdl.handle.net/2117/2499420190701T07:48:32Z20141211T08:09:29ZAlgebraic description of Jacobians isogeneous to certain Prym varieties with polarization (1,2)
Fedorov, Yuri; Enolski, Viktor Z.
For a class of nonhyperelliptic genus 3 curves C which are 2fold coverings of elliptic curves E, we give an explicit algebraic description of all birationally nonequivalent genus 2 curves whose Jacobians are degree 2 isogeneous to the Prym varieties associated to such coverings. Our description is based on previous studies of Prym varieties with polarization (1,2) in connection with separation of variables in a series of classical and new algebraic integrable systems linearized on such varieties.
We also consider some special cases of the covering C > E, in particular, when the corresponding Prym varieties contain pairs of elliptic curves and the Jacobian of C is isogeneous (but not isomorphic) to the product of 3 different elliptic curves.
Our description is accompanied with explicit numerical examples
20141211T08:09:29ZFedorov, YuriEnolski, Viktor Z.For a class of nonhyperelliptic genus 3 curves C which are 2fold coverings of elliptic curves E, we give an explicit algebraic description of all birationally nonequivalent genus 2 curves whose Jacobians are degree 2 isogeneous to the Prym varieties associated to such coverings. Our description is based on previous studies of Prym varieties with polarization (1,2) in connection with separation of variables in a series of classical and new algebraic integrable systems linearized on such varieties.
We also consider some special cases of the covering C > E, in particular, when the corresponding Prym varieties contain pairs of elliptic curves and the Jacobian of C is isogeneous (but not isomorphic) to the product of 3 different elliptic curves.
Our description is accompanied with explicit numerical examplesRigidity of Poisson Lie group actionsMiranda Galcerán, Evahttp://hdl.handle.net/2117/2463220190124T10:41:50Z20141110T12:51:24ZRigidity of Poisson Lie group actions
Miranda Galcerán, Eva
n this paper we prove that close infinitesimal momentum maps associated to Poisson Lie actions are equivalent under some mild assumptions. We also obtain rigidity theorems for actual momentum maps (when the acting Lie group G is endowed with an arbitrary Poisson structure) combining a rigidity result for canonical Hamiltonian actions (\cite{MMZ}) and a linearization theorem(\cite{GW}). These results have applications to quantization of symmetries since these infinitesimal momentum maps appear as the classical limit of quantum momentum maps (\cite{BEN}).
20141110T12:51:24ZMiranda Galcerán, Evan this paper we prove that close infinitesimal momentum maps associated to Poisson Lie actions are equivalent under some mild assumptions. We also obtain rigidity theorems for actual momentum maps (when the acting Lie group G is endowed with an arbitrary Poisson structure) combining a rigidity result for canonical Hamiltonian actions (\cite{MMZ}) and a linearization theorem(\cite{GW}). These results have applications to quantization of symmetries since these infinitesimal momentum maps appear as the classical limit of quantum momentum maps (\cite{BEN}).Lyubeznik numbers of local rings and linear strands of graded idealsÁlvarez Montaner, JosepYanagawa, Kohjihttp://hdl.handle.net/2117/2418520190124T10:41:24Z20140930T09:46:24ZLyubeznik numbers of local rings and linear strands of graded ideals
Álvarez Montaner, Josep; Yanagawa, Kohji
n this work we intro duce a new set of invariants asso ciated to the linear
strands of a minimal free resolution of a
Z
graded ideal
I
R
=

[
x
1
;:::;x
n
]
. We
also prove that these invariants satisfy some prop erties analogous to those of Lyub eznik
numb ers of lo cal rings. In particular, they satisfy a consecutiveness prop erty that we
prove rst for Lyub eznik numb ers. For the case of squarefree monomial ideals we get
more insight on the relation b etween Lyub eznik numb ers and the linear strands of their
asso ciated Alexander dual ideals. Finally, we prove that Lyub eznik numb ers of Stanley
Reisner rings are not only an algebraic invariant but also a top ological invariant, meaning
that they dep end on the homeomorphic class of the geometric realization of the asso ciated
simplicial complex and the characteristic of the base field
20140930T09:46:24ZÁlvarez Montaner, JosepYanagawa, Kohjin this work we intro duce a new set of invariants asso ciated to the linear
strands of a minimal free resolution of a
Z
graded ideal
I
R
=

[
x
1
;:::;x
n
]
. We
also prove that these invariants satisfy some prop erties analogous to those of Lyub eznik
numb ers of lo cal rings. In particular, they satisfy a consecutiveness prop erty that we
prove rst for Lyub eznik numb ers. For the case of squarefree monomial ideals we get
more insight on the relation b etween Lyub eznik numb ers and the linear strands of their
asso ciated Alexander dual ideals. Finally, we prove that Lyub eznik numb ers of Stanley
Reisner rings are not only an algebraic invariant but also a top ological invariant, meaning
that they dep end on the homeomorphic class of the geometric realization of the asso ciated
simplicial complex and the characteristic of the base fieldA methodology for obtaining asymptotic estimates for the exponentially small splitting of separatrices to whiskered tori with quadratic frequenciesDelshams Valdés, AmadeuGonchenko, MarinaGutiérrez Serrés, Perehttp://hdl.handle.net/2117/2415520190124T10:41:22Z20140925T08:12:02ZA methodology for obtaining asymptotic estimates for the exponentially small splitting of separatrices to whiskered tori with quadratic frequencies
Delshams Valdés, Amadeu; Gonchenko, Marina; Gutiérrez Serrés, Pere
20140925T08:12:02ZDelshams Valdés, AmadeuGonchenko, MarinaGutiérrez Serrés, PereContinuation of the exponentially small lower bounds for the splitting of separatrices to a whiskered torus with silver ratioDelshams Valdés, AmadeuGonchenko, MarinaGutiérrez Serrés, Perehttp://hdl.handle.net/2117/2413820190124T10:41:21Z20140923T09:33:01ZContinuation of the exponentially small lower bounds for the splitting of separatrices to a whiskered torus with silver ratio
Delshams Valdés, Amadeu; Gonchenko, Marina; Gutiérrez Serrés, Pere
We study the exponentially small splitting of invariant manifolds of whiskered (hyperbolic) tori with two fast
frequencies in nearlyintegrable Hamiltonian systems whose hyperbolic part is given by a pendulum. We consider a torus whose frequency ratio is the silver number $\Omega=\sqrt21$. We show that the oincareMelnikov 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 $\varepsilon$ satisffies a periodicity property. We also prove the continuation of the transversality of the homoclinic orbits for all the sufficiently small values of $\varepsilon
20140923T09:33:01ZDelshams Valdés, AmadeuGonchenko, MarinaGutiérrez Serrés, PereWe study the exponentially small splitting of invariant manifolds of whiskered (hyperbolic) tori with two fast
frequencies in nearlyintegrable Hamiltonian systems whose hyperbolic part is given by a pendulum. We consider a torus whose frequency ratio is the silver number $\Omega=\sqrt21$. We show that the oincareMelnikov 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 $\varepsilon$ satisffies a periodicity property. We also prove the continuation of the transversality of the homoclinic orbits for all the sufficiently small values of $\varepsilon