Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 38, 2013, 489-514
Université de Lille I, UFR de Mathématiques,
UMR 8524 du CNRS
59655 Villeneuve d'Ascq Cedex, France; volker.mayer 'at' math.univ-lille1.fr
Universidad Católica del Norte, Departamento de Matemáticas
Avenida Angamos 0610, Antofagasta, Chile; bskorulski 'at' ucn.cl
University of North Texas, Department of Mathematics
Denton, TX 76203-1430, U.S.A.; urbanski 'at' unt.edu
Abstract. This note concerns non-autonomous dynamics of rational functions and, more precisely, the fractal behavior of the Julia sets under perturbation of non-autonomous systems. We provide a necessary and sufficient condition for holomorphic stability which leads to Hölder continuity of dimensions of hyperbolic non-autonomous Julia sets with respect to the l\infty-topology on the parameter space. On the other hand we show that, for some particular family, the Hausdorff and packing dimension functions are not differentiable at any point and that these dimensions are not equal on an open dense set of the parameter space still with respect to the l\infty-topology.
2010 Mathematics Subject Classification: Primary 30D05.
Key words: Holomorphic dynamics, holomorphic motions, meromorphic functions.
Reference to this article: V. Mayer, B. Skorulski and M. Urbanski: Regularity and irregularity of fiber dimensions of non-autonomous dynamical systems. Ann. Acad. Sci. Fenn. Math. 38 (2013), 489-514.
doi:10.5186/aasfm.2013.3829
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