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- Subjects
- 37-XX Dynamical systems and ergodic theory
We study the geometry of -bundles—locally projective -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 -bundles; in particular, we
prove that the local structure of -bundles is captured by the full Sato Grassmannian. The rational, trigonometric, and elliptic solutions of KP...
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...
We study the frequency of hypercyclicity of hypercyclic, non–weakly mixing linear operators. In particular, we show that on the space , any sublinear frequency can be realized by a non–weakly mixing operator. A weaker but similar result is obtained for or , . Part of our results is related to some Sidon-type lacunarity properties for sequences of natural numbers.
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.
We give analogs of the complexity and of Sturmian words which are called respectively the -complexity and -Sturmian words. We show that the class of -Sturmian words coincides with the class of words satisfying , and we determine the structure of -Sturmian words. For a class of words satisfying , we give a general formula and an upper bound for . Using this general formula, we give explicit formulae for for some words belonging to this class. In general, can take large values, namely,...
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,...
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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