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Thermodynamical formalism and multifractal analysis for meromorphic functions of finite order
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Thermodynamical formalism and multifractal analysis for meromorphic functions of finite order
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eBook
Thermodynamical formalism and multifractal analysis for meromorphic functions of finite order
2010
Overview
The thermodynamical formalism has been developed by the authors for a very general class of transcendental meromorphic functions. A
function
In the present manuscript we first improve upon our earlier paper
in providing a systematic account of the thermodynamical formalism for such a meromorphic function
Then we provide various, mainly geometric, applications
of this theory. Indeed, we examine the finer fractal structure of the radial (in fact non-escaping) Julia set by developing the
multifractal analysis of Gibbs states. In particular, the Bowen’s formula for the Hausdorff dimension of the radial Julia set from our
earlier paper is reproved. Moreover, the multifractal spectrum function is proved to be convex, real-analytic and to be the Legendre
transform conjugate to the temperature function. In the last chapter we went even further by showing that, for a analytic family
satisfying a symmetric version of the growth condition (1.1) in a uniform way, the multifractal spectrum function is real-analytic also
with respect to the parameter. Such a fact, up to our knowledge, has not been so far proved even for hyperbolic rational functions nor
even for the quadratic family
Publisher
American Mathematical Society
ISBN
9780821846599, 0821846590
