Boundedness of hyperbolic components of Newton maps
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BOUNDEDNESS OF HYPERBOLIC COMPONENTS OF NEWTON MAPS BY
Hongming Nie Einstein Institute of Mathematics, The Hebrew University of Jerusalem Givat Ram, Jerusalem 91904, Israel e-mail: [email protected] AND
Kevin M. Pilgrim Department of Mathematics, Indiana University, Bloomington, Indiana 47405, USA e-mail: [email protected]
ABSTRACT
We investigate boundedness of hyperbolic components in the moduli space of Newton maps. For quartic maps, (i) we prove hyperbolic components possessing two distinct attracting cycles each of period at least two are bounded, and (ii) we characterize the possible points on the boundary at infinity for some other types of hyperbolic components. For general maps, we prove hyperbolic components whose elements have fixed superattracting basins mapping by degree at least three are unbounded.
1. Introduction In this paper, we study boundedness properties of hyperbolic components of Newton maps. A rational map is hyperbolic if every critical point converges to a (super)attracting cycle under iteration. Hyperbolicity is an open condition invariant under conjugacy. Conjecturally hyperbolic maps are dense in moduli space; see [22, 26]. A connected component of the set of the hyperbolic maps is called Received November 1, 2018 and in revised form August 19, 2019
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H. NIE AND K. M. PILGRIM
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a hyperbolic component. Any two maps in the same hyperbolic component are quasiconformally conjugate on a neighborhood of their Julia sets [15]. For a hyperbolic component whose elements have connected Julia sets, there is up to M¨obius conjugacy a unique post-critically finite map in this component [24]. In terms of the “dictionary” between rational maps and Kleinian groups, hyperbolic components are analogous to components of the set of convex compact discrete faithful representations of a fixed group into PSL2 (C); see [24, 37]. For more properties of hyperbolic components, we refer to [28, 38]. Our general focus is on hyperbolic components in algebraic subfamilies F of the moduli space ratd of rational maps of degree d, which is the quotient of the parameter space Ratd ⊂ P2d+1 of all rational maps of degree d under the conjugation action by Aut(P1 ). If H ⊂ F ⊂ ratd is a component of hyperbolic maps in F, we say it is bounded in F if the closure H ⊂ ratd is compact. The simplest examples of hyperbolic components arise in the quadratic polynomial family Pc (z) = z 2 + c. In this case, there two kinds of components. The unbounded escape locus is the complement of the Mandelbrot set, and has fractal boundary. The bounded components are uniformized by the multiplier λ of the unique attracting cycle [10], so as subsets of the plane they are semi-algebraic, defined as a component of the real-algebraic inequality |λ| < 1. More general structural results regarding hyperbolic components in the spaces of higher degree polynomials are given in [32, 39, 44]. Boundedness results. For quadratic rational maps, in [38] Rees divided the hyperbolic components into 4 types (see also [29]), and show
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