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Zbl 0937.37025
Sester, Olivier
Hyperbolicité des polynômes fibrés. (Hyperbolicity of fibered polynomials). (French)
[J] Bull. Soc. Math. Fr. 127, No.3, 393-428 (1999). ISSN 0037-9484

The author studies fibred polynomials $(x,z)\mapsto (f(x), c_d(x) z^d+\dots+ c_0(x))$, where $f:X\to X$ is a self-map of a compact space $X$ and the $c_0,\dots, c_d$ $(d\geq 2)$ are continuous maps of $X$ in $\bbfC$. He gives a condition which ensures that the action on the Julia sets in the fibres (defined by fibre-wise iteration) is expanding. He then discusses the special case of $d=2$ where he assumes in addition that these Julia sets are quasi-circles. The latter case (for holomorphic $f$ but without the additional assumption) is already discussed by {\it S. M. Heinemann} [Kyushu J. Math. 52, 299-329 (1998; Zbl 0931.37015)] in the broader context of Julia sets of higher-dimensional spaces, which generalizes to the setting of fibred polynomials.
[Manfred Denker (Göttingen)]

MSC 2000:: *37F15 Expanding maps; hyperbolicity; structural stability
37H05 Foundations, general theory of cocycles, algebraic ergodic theory

Keywords: fibered polynomial; holomorphic dynamics; Julia set; Green function; fibered complex polynomials; expanding; Julia sets; quasi-circles; fibred polynomials

Citations: Zbl 0931.37015

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