2,779 research outputs found
Guest Post: John Rambo fights the Civil War
Today we offer another of our “Guest Post Wednesday” extras, with some meditations on Rambo from Aaron Urbanski. Yeah, we’re not kidding... Rambo. Aaron is a graduate of the GMU American History Masters program and alumnus Ranger of the NPS. [excerpt
Geometric Thermodynamical Formalism and Real Analyticity for Meromorphic Functions of Finite Order
Working with well chosen Riemannian metrics and employing Nevanlinna's
theory, we make the thermodynamical formalism work for a wide class of
hyperbolic meromorphic functions of finite order (including in particular
exponential family, elliptic functions, cosine, tangent and the cosine--root
family and also compositions of these functions with arbitrary polynomials). In
particular, the existence of conformal (Gibbs) measures is established and then
the existence of probability invariant measures equivalent to conformal
measures is proven. As a geometric consequence of the developed thermodynamic
formalism, a version of Bowen's formula expressing the Hausdorff dimension of
the radial Julia set as the zero of the pressure function and, moreover, the
real analyticity of this dimension, is proved.Comment: 32 page
On characterization of Poisson and Jacobi structures
We characterize Poisson and Jacobi structures by means of complete lifts of
the corresponding tensors: the lifts have to be related to canonical structures
by morphisms of corresponding vector bundles. Similar results hold for
generalized Poisson and Jacobi structures (canonical structures) associated
with Lie algebroids and Jacobi algebroids.Comment: LaTeX, 14 page
Random Dynamics of Transcendental Functions
This work concerns random dynamics of hyperbolic entire and meromorphic
functions of finite order and whose derivative satisfies some growth condition
at infinity. This class contains most of the classical families of
transcendental functions and goes much beyond. Based on uniform versions of
Nevanlinna's value distribution theory we first build a thermodynamical
formalism which, in particular, produces unique geometric and fiberwise
invariant Gibbs states. Moreover, spectral gap property for the associated
transfer operator along with exponential decay of correlations and a central
limit theorem are shown. This part relies on our construction of new positive
invariant cones that are adapted to the setting of unbounded phase spaces. This
setting rules out the use of Hilbert's metric along with the usual contraction
principle. However these cones allow us to apply a contraction argument
stemming from Bowen's initial approach.Comment: Final Version, to appear in J. d'Analyse Math, 35 page
Real analyticity of Hausdorff dimension for expanding rational semigroups
We consider the dynamics of expanding semigroups generated by finitely many
rational maps on the Riemann sphere. We show that for an analytic family of
such semigroups, the Bowen parameter function is real-analytic and
plurisubharmonic. Combining this with a result obtained by the first author, we
show that if for each semigroup of such an analytic family of expanding
semigroups satisfies the open set condition, then the function of the Hausdorff
dimension of the Julia set is real-analytic and plurisubharmonic. Moreover, we
provide an extensive collection of classes of examples of analytic families of
semigroups satisfying all the above conditions and we analyze in detail the
corresponding Bowen's parameters and Hausdorff dimension function.Comment: 33 pages, 2 figures. Some typos are fixed. Published in Ergodic
Theory Dynam. Systems (2010), Vol. 30, No. 2, 601-633
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