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Desingularizing -symplectic structures
A -dimensional Poisson manifold is said to be -symplectic
if it is symplectic on the complement of a hypersurface and has a simple
Darboux canonical form at points of which we will describe below. In this
paper we will discuss a desingularization procedure which, for even,
converts into a family of symplectic forms having the
property that is equal to the -symplectic form dual to
outside an -neighborhood of and, in addition, converges to
this form as tends to zero in a sense that will be made precise in
the theorem below. We will then use this construction to show that a number of
somewhat mysterious properties of -manifolds can be more clearly
understood by viewing them as limits of analogous properties of the
's. We will also prove versions of these results for
odd; however, in the odd case the family has to be replaced
by a family of folded symplectic forms.Comment: new version, 13 pages, 3 figures, final version accepted at IMRN,
International Mathematics Research Notice
Generalized Thue-Morse words and palindromic richness
We prove that the generalized Thue-Morse word defined for
and as , where denotes the sum of digits in the base-
representation of the integer , has its language closed under all elements
of a group isomorphic to the dihedral group of order consisting of
morphisms and antimorphisms. Considering simultaneously antimorphisms , we show that is saturated by -palindromes
up to the highest possible level. Using the terminology generalizing the notion
of palindromic richness for more antimorphisms recently introduced by the
author and E. Pelantov\'a, we show that is -rich. We
also calculate the factor complexity of .Comment: 11 page
An Invitation to Singular Symplectic Geometry
In this paper we analyze in detail a collection of motivating examples to
consider -symplectic forms and folded-type symplectic structures. In
particular, we provide models in Celestial Mechanics for every -symplectic
structure. At the end of the paper, we introduce the odd-dimensional analogue
to -symplectic manifolds: -contact manifolds.Comment: 14 pages, 1 figur
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