439 research outputs found
Logarithm laws and shrinking target properties
We survey some of the recent developments in the study of logarithm laws and
shrinking target properties for various families of dynamical systems. We
discuss connections to geometry, diophantine approximation, and probability
theory.Comment: This is a survey paper written following the Conference on Measures
and Dyanmics on groups and homogeneous spaces, at TIFR, Mumbai, in Dec. 2007.
It is in honor of Prof. S.G. Dani's 60th Birthda
Counting generalized Jenkins-Strebel differentials
We study the combinatorial geometry of "lattice" Jenkins--Strebel
differentials with simple zeroes and simple poles on and of the
corresponding counting functions. Developing the results of M. Kontsevich we
evaluate the leading term of the symmetric polynomial counting the number of
such "lattice" Jenkins-Strebel differentials having all zeroes on a single
singular layer. This allows us to express the number of general "lattice"
Jenkins-Strebel differentials as an appropriate weighted sum over decorated
trees.
The problem of counting Jenkins-Strebel differentials is equivalent to the
problem of counting pillowcase covers, which serve as integer points in
appropriate local coordinates on strata of moduli spaces of meromorphic
quadratic differentials. This allows us to relate our counting problem to
calculations of volumes of these strata . A very explicit expression for the
volume of any stratum of meromorphic quadratic differentials recently obtained
by the authors leads to an interesting combinatorial identity for our sums over
trees.Comment: to appear in Geometriae Dedicata. arXiv admin note: text overlap with
arXiv:1212.166
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