Topologies on the set of iterates of a holomorphic function in infinite dimensions

• Autorzy: Mackey M. , Mellon P.

Identyfikatory

DOI

10.4064/ba220731-22-12

• Języki publikacji: EN

Abstrakty

EN

Let f: B→B be a compact holomorphic map on the open unit ball B of a complex Banach space Z in possibly infinite dimensions, where f compact means f(B) is relatively compact. The sequence of iterates (fn)n of f (where fn:=f∘fn−1, f1:=f) is of much interest and, since it generally does not converge, the set of all its subsequential limits for a particular topology have been studied instead. We prove that the pointwise limit of any subsequence of (fn)n is itself a holomorphic function. We show, in fact, that on the set of iterates {fn : n∈N} the topology of pointwise convergence on B coincides with any finer topology on the space H(B,Z) of holomorphic functions from B to Z. In particular, it coincides with both the compact-open topology and the topology of local uniform convergence on B. Despite the fact that these topologies are not first countable, we prove that the set of accumulation points of (fn)n coincides with the set of all its subsequential limits.

Słowa kluczowe

EN

holomorphic mappings; iteration; convergence of iterates; subsequential limits

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2022
• Tom: Vol. 70, no. 2
• Strony: 151--158
• Opis fizyczny: Bibliogr. 24 poz.

Twórcy

autor

Mackey M.

• School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland

• https://orcid.org/0000-0002-1936-1821

autor

Mellon P.

• School of Mathematics and Statistics University College Dublin Belfield, Dublin 4, Ireland

• https://orcid.org/0000-0003-1375-8462

Bibliografia

• [1] M. Abate, Horospheres and iterates of holomorphic maps, Math. Z. 198 (1988), 225-238.
• [2] M. Abate, Iteration theory, compactly divergent sequences and commuting holomorphic maps, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 18 (1991), 167-191.
• [3] M. Abate and J. Raissy, Wolff-Denjoy theorems in nonsmooth convex domains, Ann. Mat. Pura Appl. (4) 193 (2014), 1503-1518.
• [4] L. Arosio and F. Bracci, Canonical models for holomorphic iteration, Trans. Amer. Math. Soc. 368 (2016), 3305-3339.
• [5] M. Budzyńska, T. Kuczumow and S. Reich, A Denjoy-Wolff theorem for compact holomorphic mappings in reflexive Banach spaces, J. Math. Anal. Appl. 396 (2012), 504-512.
• [6] M. Budzyńska, T. Kuczumow and S. Reich, A Denjoy-Wolf theorem for compact holomorphic mappings in complex Banach spaces, Ann. Acad. Sci. Fenn. Math. 38 (2013), 747-756.
• [7] C. H. Chu and P. Mellon, Iteration of compact holomorphic maps on a Hilbert ball, Proc. Amer. Math. Soc. 125 (1997), 1771-1777.
• [8] A. Denjoy, Sur l’itération des fonctions analytiques, C. R. Acad. Sci. Paris 182 (1926), 255-257.
• [9] K. Goebel and S. Reich, Iterating holomorphic self-mappings of the Hilbert ball, Proc. Japan Acad. 58 (1982), 349-352.
• [10] K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry and Non-expansive Mappings, Dekker, 1984.
• [11] M. Hervé, Itération des transformations analytiques dans le bicercle-unité, Ann. Sci. École Norm. Sup. (3) 71 (1954), 1-28.
• [12] M. Hervé, Quelques propriétés des applications analytiques d’une boule ŕ m dimensions dans elle-męme, J. Math. Pures Appl. 42 (1963), 117-147.
• [13] G. Julia, Mémoire sur l’itération des fonctions rationnelles, J. Math. Pures Appl. 8 (1918), 47-245.
• [14] J. Kapeluszny, T. Kuczumow and S. Reich, The Denjoy-Wolff theorem in the open unit ball of a strictly convex Banach space, Adv. Math. 143 (1999), 111-123.
• [15] A. Kryczka and T. Kuczumow, The Denjoy-Wolff-type theorem for compact kBH-nonexpansive maps on a Hilbert ball, in: Proceedings of Workshop on Fixed Point Theory (Kazimierz Dolny, 1997), Ann. Univ. Mariae Curie-Skłodowska Sect. A 51 (1997), 179-183.
• [16] M. Mackey and P. Mellon, Iterates of a compact holomorphic map on a finite rank homogeneous ball, Acta Sci. Math. (Szeged) 85 (2019), 203-214.
• [17] P. Mellon, Holomorphic invariance on bounded symmetric domains, J. Reine Angew. Math. 523 (2000), 199-223.
• [18] P. Mellon, Dynamics of biholomorphic self-maps on bounded symmetric domains, Math. Scand. 117 (2015), 203-216.
• [19] P. Mellon, Denjoy-Wolff theory for finite-dimensional bounded symmetric domains, Ann. Mat. Pura Appl. (4) 195 (2016), 845-855.
• [20] A. Stachura, Iterations of holomorphic self-maps of the unit ball in Hilbert space, Proc. Amer. Math. Soc. 93 (1985), 88-90.
• [21] E. Thorp and R. Whitley, The strong maximum modulus theorem for analytic functions into a Banach space, Proc. Amer. Math. Soc. 18 (1967), 640-646.
• [22] J. Wolff, Sur l’itération des fonctions bornées, C. R. Acad. Sci. Paris 182 (1926), 200-201.
• [23] J. Wolff, Sur une généralisation d’un théorčme de Schwartz (sic!), C. R. Acad. Sci. Paris 183 (1926), 500-502.
• [24] J. Wolff, Sur une généralisation d’un théorčme de Schwarz, C. R. Acad. Sci. Paris 182 (1926), 918-920.

Uwagi

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).