151 research outputs found
Dynamical degrees of birational transformations of projective surfaces
The dynamical degree of a birational transformation measures
the exponential growth rate of the degree of the formulae that define the
-th iterate of . We study the set of all dynamical degrees of all
birational transformations of projective surfaces, and the relationship between
the value of and the structure of the conjugacy class of . For
instance, the set of all dynamical degrees of birational transformations of the
complex projective plane is a closed and well ordered set of algebraic numbers.Comment: 65 page
Regularization by free additive convolution, square and rectangular cases
The free convolution is the binary operation on the set of probability
measures on the real line which allows to deduce, from the individual spectral
distributions, the spectral distribution of a sum of independent unitarily
invariant square random matrices or of a sum of free operators in a non
commutative probability space. In the same way, the rectangular free
convolution allows to deduce, from the individual singular distributions, the
singular distribution of a sum of independent unitarily invariant rectangular
random matrices. In this paper, we consider the regularization properties of
these free convolutions on the whole real line. More specifically, we try to
find continuous semigroups of probability measures such that
is the Dirac mass at zero and such that for all positive and all
probability measure , the free convolution of with (or, in
the rectangular context, the rectangular free convolution of with
) is absolutely continuous with respect to the Lebesgue measure, with a
positive analytic density on the whole real line. In the square case, we prove
that in semigroups satisfying this property, no measure can have a finite
second moment, and we give a sufficient condition on semigroups to satisfy this
property, with examples. In the rectangular case, we prove that in most cases,
for in a continuous rectangular-convolution-semigroup, the rectangular
convolution of with either has an atom at the origin or doesn't put
any mass in a neighborhood of the origin, thus the expected property does not
hold. However, we give sufficient conditions for analyticity of the density of
the rectangular convolution of with except on a negligible set of
points, as well as existence and continuity of a density everywhere.Comment: 43 pages, to appear in Complex Analysis and Operator Theor
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