381,477 research outputs found
Large Bergman spaces: invertibility, cyclicity, and subspaces of arbitrary index
In a wide class of weighted Bergman spaces, we construct invertible
non-cyclic elements. These are then used to produce z-invariant subspaces of
index higher than one. In addition, these elements generate nontrivial
bilaterally invariant subspaces in anti-symmetrically weighted Hilbert spaces
of sequences.Comment: 40 page
Fractal Strings and Multifractal Zeta Functions
For a Borel measure on the unit interval and a sequence of scales that tend
to zero, we define a one-parameter family of zeta functions called multifractal
zeta functions. These functions are a first attempt to associate a zeta
function to certain multifractal measures. However, we primarily show that they
associate a new zeta function, the topological zeta function, to a fractal
string in order to take into account the topology of its fractal boundary. This
expands upon the geometric information garnered by the traditional geometric
zeta function of a fractal string in the theory of complex dimensions. In
particular, one can distinguish between a fractal string whose boundary is the
classical Cantor set, and one whose boundary has a single limit point but has
the same sequence of lengths as the complement of the Cantor set. Later work
will address related, but somewhat different, approaches to multifractals
themselves, via zeta functions, partly motivated by the present paper.Comment: 32 pages, 9 figures. This revised version contains new sections and
figures illustrating the main results of this paper and recent results from
others. Sections 0, 2, and 6 have been significantly rewritte
Rigidity of escaping dynamics for transcendental entire functions
We prove an analog of Boettcher's theorem for transcendental entire functions
in the Eremenko-Lyubich class B. More precisely, let f and g be entire
functions with bounded sets of singular values and suppose that f and g belong
to the same parameter space (i.e., are *quasiconformally equivalent* in the
sense of Eremenko and Lyubich). Then f and g are conjugate when restricted to
the set of points which remain in some sufficiently small neighborhood of
infinity under iteration. Furthermore, this conjugacy extends to a
quasiconformal self-map of the plane.
We also prove that this conjugacy is essentially unique. In particular, we
show that an Eremenko-Lyubich class function f has no invariant line fields on
its escaping set.
Finally, we show that any two hyperbolic Eremenko-Lyubich class functions f
and g which belong to the same parameter space are conjugate on their sets of
escaping points.Comment: 28 pages; 2 figures. Final version (October 2008). Various
modificiations were made, including the introduction of Proposition 3.6,
which was not formally stated previously, and the inclusion of a new figure.
No major changes otherwis
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