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𝒟 -bundles and integrable hierarchies

David Ben-Zvi, Thomas Nevins (2011)

Journal of the European Mathematical Society

We study the geometry of 𝒟 -bundles—locally projective 𝒟 -modules—on algebraic curves, and apply them to the study of integrable hierarchies, specifically the multicomponent Kadomtsev–Petviashvili (KP) and spin Calogero–Moser (CM) hierarchies. We show that KP hierarchies have a geometric description as flows on moduli spaces of 𝒟 -bundles; in particular, we prove that the local structure of 𝒟 -bundles is captured by the full Sato Grassmannian. The rational, trigonometric, and elliptic solutions of KP...

∞-jets of diffeomorphisms preserving orbits of vector fields

Sergiy Maksymenko (2009)

Open Mathematics

Let F be a C ∞ vector field defined near the origin O ∈ ℝn, F(O) = 0, and (Ft) be its local flow. Denote by the set of germs of orbit preserving diffeomorphisms h: ℝn → ℝn at O, and let , (r ≥ 0), be the identity component of with respect to the weak Whitney Wr topology. Then contains a subset consisting of maps of the form Fα(x)(x), where α: ℝn → ℝ runs over the space of all smooth germs at O. It was proved earlier by the author that if F is a linear vector field, then = . In this paper we present...

(Non-)weakly mixing operators and hypercyclicity sets

Frédéric Bayart, Étienne Matheron (2009)

Annales de l’institut Fourier

We study the frequency of hypercyclicity of hypercyclic, non–weakly mixing linear operators. In particular, we show that on the space ℓ 1 ( ℕ ) , any sublinear frequency can be realized by a non–weakly mixing operator. A weaker but similar result is obtained for c 0 ( ℕ ) or ℓ p ( ℕ ) , 1 < p < ∞ . Part of our results is related to some Sidon-type lacunarity properties for sequences of natural numbers.

⊗-product of Markov matrices.

J. P. Lampreia, A. Rica da Silva, J. Sousa Ramos (1988)

Stochastica

In this paper we introduce a ⊗-operation over Markov transition matrices, in the context of subshift of finite type, reproducing symbolic properties of the iterates of the critical point on a one-parameter family of unimodal maps. To the *-product between kneading sequences we associate a ⊗-product between the corresponding Markov matrices.

*-sturmian words and complexity

Izumi Nakashima, Jun-Ichi Tamura, Shin-Ichi Yasutomi (2003)

Journal de théorie des nombres de Bordeaux

We give analogs of the complexity p ( n ) and of Sturmian words which are called respectively the * -complexity p * ( n ) and * -Sturmian words. We show that the class of * -Sturmian words coincides with the class of words satisfying p * ( n ) ≤ n + 1 , and we determine the structure of * -Sturmian words. For a class of words satisfying p * ( n ) = n + 1 , we give a general formula and an upper bound for p ( n ) . Using this general formula, we give explicit formulae for p ( n ) for some words belonging to this class. In general, p ( n ) can take large values, namely,...

[unknown]

Takato Uehara (0)

Annales de l’institut Fourier

[unknown]

Semyon Dyatlov (0)

Annales de l’institut Fourier

[unknown]

Sébastien Alvarez, Nicolas Hussenot (0)

Annales de l’institut Fourier

[unknown]

Matthias Leuenberger (0)

Annales de l’institut Fourier

ω-Limit sets for triangular mappings

Victor Jiménez López, Jaroslav Smítal (2001)

Fundamenta Mathematicae

In 1992 Agronsky and Ceder proved that any finite collection of non-degenerate Peano continua in the unit square is an ω-limit set for a continuous map. We improve this result by showing that it is valid, with natural restrictions, for the triangular maps (x,y) ↦ (f(x),g(x,y)) of the square. For example, we show that a non-trivial Peano continuum C ⊂ I² is an orbit-enclosing ω-limit set of a triangular map if and only if it has a projection property. If C is a finite union of Peano continua then,...

Ω-stability for maps with nonwandering critical points

J. Delgado, N. Romero, A. Rovella, F. Vilamajó (2007)

Fundamenta Mathematicae

Sufficient conditions for a map having nonwandering critical points to be Ω-stable are introduced. It is not known if these conditions are necessary, but they are easily verified for all known examples of Ω-stable maps. Their necessity is shown in dimension two. Examples are given of Axiom A maps that have no cycles but are not Ω-stable.

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