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Conformal Geometry and Dynamics

ISSN 1088-4173

 

 

Dynamical properties of families of holomorphic mappings


Authors: Ratna Pal and Kaushal Verma
Journal: Conform. Geom. Dyn. 19 (2015), 323-350
MSC (2010): Primary 37F45
DOI: https://doi.org/10.1090/ecgd/285
Published electronically: December 10, 2015
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Abstract: We study some dynamical properties of skew products of Hénon maps of $ \mathbb{C}^2$ that are fibered over a compact metric space $ M$. The problem reduces to understanding the dynamical behavior of the composition of a pseudo-random sequence of Hénon mappings. In analogy with the dynamics of the iterates of a single Hénon map, it is possible to construct fibered Green functions that satisfy suitable invariance properties and the corresponding stable and unstable currents. This analogy is carried forth in two ways: it is shown that the successive pull-backs of a suitable current by the skew Hénon maps converges to a multiple of the fibered stable current and secondly, this convergence result is used to obtain a lower bound on the topological entropy of the skew product in some special cases. The other class of maps that are studied are skew products of holomorphic endomorphisms of $ \mathbb{P}^k$ that are again fibered over a compact base. We define the fibered Fatou component and show that they are pseudoconvex and Kobayashi hyperbolic.


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Additional Information

Ratna Pal
Affiliation: Department of Mathematics, Indian Institute of Science, Bangalore 560 012, India
Email: ratna10@math.iisc.ernet.in

Kaushal Verma
Affiliation: Department of Mathematics, Indian Institute of Science, Bangalore 560 012, India
Email: kverma@math.iisc.ernet.in

DOI: https://doi.org/10.1090/ecgd/285
Received by editor(s): April 22, 2015
Received by editor(s) in revised form: September 28, 2015
Published electronically: December 10, 2015
Additional Notes: The first named author was supported by CSIR-UGC (India) fellowship
The second named author was supported by the DST Swarna Jayanti Fellowship 2009–2010 and a UGC–CAS Grant
Article copyright: © Copyright 2015 American Mathematical Society