Abstract. We consider holomorphic self-maps f of the unit ball B^N in C^N (N = 1, 2, 3, . . . ). In the one-dimensional case, when f has no fixed points in D := B^1 and is of hyperbolic type, there is a classical renormalization procedure due to Valiron which allows to semi-linearize the map f, and therefore, in this case, the dynamical properties of f are well understood. In this paper, we generalize the classical Valiron construction to higher dimensions under some weak assumptions on f at its Denjoy-Wolff point. As a result, we construct a semiconjugation h, which maps the ball into the right half-plane of C, and solves the functional equation h(f(z))= rh(z), where r > 1 is the (inverse of the) boundary dilation coefficient at the Denjoy-Wolff point of f.
Valiron's construction in higher dimension / Bracci, Filippo; Gentili, Graziano; Poggi-Corradini, Pietro. - In: REVISTA MATEMATICA IBEROAMERICANA. - ISSN 0213-2230. - STAMPA. - 26 (1):(2010), pp. 57-76. [10.4171/RMI/593]
Valiron's construction in higher dimension.
GENTILI, GRAZIANO;
2010
Abstract
Abstract. We consider holomorphic self-maps f of the unit ball B^N in C^N (N = 1, 2, 3, . . . ). In the one-dimensional case, when f has no fixed points in D := B^1 and is of hyperbolic type, there is a classical renormalization procedure due to Valiron which allows to semi-linearize the map f, and therefore, in this case, the dynamical properties of f are well understood. In this paper, we generalize the classical Valiron construction to higher dimensions under some weak assumptions on f at its Denjoy-Wolff point. As a result, we construct a semiconjugation h, which maps the ball into the right half-plane of C, and solves the functional equation h(f(z))= rh(z), where r > 1 is the (inverse of the) boundary dilation coefficient at the Denjoy-Wolff point of f.File | Dimensione | Formato | |
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