Backward dynamics of non-expanding maps in Gromov hyperbolic metric spaces.(English)Zbl 1545.32043
The authors study the interplay between the backward dynamics of a non-expanding self-map \(f\) of a proper geodesic Gromov hyperbolic metric space \(X\) and the boundary regular fixed points of \(f\) in the Gromov boundary as defined in [the first author et al., Math. Ann. 388, No. 2, 1163–1204 (2024;Zbl 1537.32030)].
They introduce the notion of stable dilation at a boundary regular fixed point of the Gromov boundary, whose value is related to the dynamical behaviour of the fixed point. This geometric theory applies in particular to holomorphic self-maps of bounded domains \(\Omega\subset\subset\mathbb{C}^q\), where \(\Omega\) is either strongly pseudoconvex, smoothly bounded convex of finite D’Angelo type, smoothly bounded pseudoconvex of finite D’Angelo type when \(q = 2\), solving several open problems from the literature.
For an analytic self-map \(f\) of the unit disk in \(\mathbb{C}\), the dynamics of forward iterates has been well understood, with the Wolff-Denjoy theorem as the main tool. The backward dynamics of \(f\) is described by two main results, obtained by Bracci and Poggi-Corradini. The first one shows the existence of backward orbits with bounded step converging to a given repelling boundary regular fixed point. The second one can be thought of as a backward version of the Wolff-Denjoy theorem. In this paper the authors extend such results and show that, like in the forward dynamics case, the holomorphic structure does not play a relevant role in the above mentioned results. More precisely, they introduce the notions of dilation and boundary regular fixed point in this setting, and they generalize both results to the case of a non-expanding self-map \(f \colon X \to X\) of a proper geodesic Gromov hyperbolic metric space. Moreover, the authors are able to prove that for holomorphic parabolic self-maps any escaping (i.e., leaving any given compact set in \(X\)) backward orbit with bounded step always converges to a point in the boundary. This was an open question so far, even for the unit ball in \(\mathbb{C}^q\).
They introduce the notion of stable dilation at a boundary regular fixed point of the Gromov boundary, whose value is related to the dynamical behaviour of the fixed point. This geometric theory applies in particular to holomorphic self-maps of bounded domains \(\Omega\subset\subset\mathbb{C}^q\), where \(\Omega\) is either strongly pseudoconvex, smoothly bounded convex of finite D’Angelo type, smoothly bounded pseudoconvex of finite D’Angelo type when \(q = 2\), solving several open problems from the literature.
For an analytic self-map \(f\) of the unit disk in \(\mathbb{C}\), the dynamics of forward iterates has been well understood, with the Wolff-Denjoy theorem as the main tool. The backward dynamics of \(f\) is described by two main results, obtained by Bracci and Poggi-Corradini. The first one shows the existence of backward orbits with bounded step converging to a given repelling boundary regular fixed point. The second one can be thought of as a backward version of the Wolff-Denjoy theorem. In this paper the authors extend such results and show that, like in the forward dynamics case, the holomorphic structure does not play a relevant role in the above mentioned results. More precisely, they introduce the notions of dilation and boundary regular fixed point in this setting, and they generalize both results to the case of a non-expanding self-map \(f \colon X \to X\) of a proper geodesic Gromov hyperbolic metric space. Moreover, the authors are able to prove that for holomorphic parabolic self-maps any escaping (i.e., leaving any given compact set in \(X\)) backward orbit with bounded step always converges to a point in the boundary. This was an open question so far, even for the unit ball in \(\mathbb{C}^q\).
Reviewer: Jasmin Raissy (Talence)
MSC:
| 32H50 | Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables |
| 32F45 | Invariant metrics and pseudodistances in several complex variables |
| 53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |
Keywords:
holomorphic dynamics;non-expanding maps;Gromov hyperbolicity;horofunctions;boundary fixed pointsCitations:
Zbl 1537.32030Cite
References:
| [1] | Abate, M., Iteration Theory of Holomorphic Maps on Taut Manifolds (1989), Mediterranean Press: Mediterranean Press Cosenza ·Zbl 0747.32002 |
| [2] | Abate, M., Horospheres and iterates of holomorphic maps, Math. Z., 198, 2, 225-238 (1988) ·Zbl 0628.32035 |
| [3] | Abate, M.; Bracci, F., Common boundary regular fixed points for holomorphic semigroups in strongly convex domains, Contemp. Math., 667, 1-14 (2016) ·Zbl 1353.32010 |
| [4] | Abate, M.; Raissy, J., Backward iteration in strongly convex domains, Adv. Math., 228, 5, 2837-2854 (2011) ·Zbl 1250.32019 |
| [5] | Abate, M.; Raissy, J., Backward iteration in strongly convex domains - Errata Corrige, Adv. Math., 369, 1-9 (2020) ·Zbl 1441.32008 |
| [6] | Altavilla, A.; Arosio, L.; Guerini, L., Canonical models on strongly convex domains via the squeezing function, J. Geom. Anal., 31, 5, 4661-4702 (2021) ·Zbl 1464.32025 |
| [7] | Arosio, L., Canonical models for the forward and backward iteration of holomorphic maps, J. Geom. Anal., 27, 2, 1178-1210 (2017) ·Zbl 1382.32012 |
| [8] | Arosio, L.; Fiacchi, M.; Gontard, S.; Guerini, L., The horofunction boundary of a Gromov hyperbolic space (2020) |
| [9] | Arosio, L.; Guerini, L., Backward orbits in the unit ball, Proc. Am. Math. Soc., 147, 9, 3947-3954 (2019) ·Zbl 1429.32024 |
| [10] | Balogh, Z. M.; Bonk, M., Gromov hyperbolicity and the Kobayashi metric on strictly pseudoconvex domains, Comment. Math. Helv., 75, 504-533 (2000) ·Zbl 0986.32012 |
| [11] | Beardon, A. F., The dynamics of contractions, Ergod. Theory Dyn. Syst., 17, 6, 1257-1266 (1997) ·Zbl 0952.54023 |
| [12] | Bedford, E., On the automorphism group of a Stein manifold, Math. Ann., 266, 215-227 (1983) ·Zbl 0532.32014 |
| [13] | Benoist, Y., Convexes hyperboliques et fonctions quasisymétriques, Publ. Math. Inst. Hautes Études Sci., 7, 181-237 (2003) ·Zbl 1049.53027 |
| [14] | Bracci, F., Fixed points of commuting holomorphic mappings other than the Wolff point, Trans. Am. Math. Soc., 355, 6, 2569-2584 (2003) ·Zbl 1045.30014 |
| [15] | Bridson, M.; Haefliger, A., Metric Spaces of Nonpositive Curvature, Grundlehren der Mathematischen Wissenschaften, vol. 319 (1999), Springer-Verlag: Springer-Verlag Berlin ·Zbl 0988.53001 |
| [16] | Coornaert, M.; Delzant, T.; Papadopoulos, A., Géométrie et théorie des groupes, Lect. Notes Math., 1441 (1990) ·Zbl 0727.20018 |
| [17] | Calka, A., On conditions under which isometries have bounded orbits, Colloq. Math., 48, 2, 219-227 (1984) ·Zbl 0558.54021 |
| [18] | Denjoy, A., Sur l’itération des fonctions analytiques, C. R. Math. Acad. Sci. Paris, 182, 255-257 (1926) ·JFM 52.0309.04 |
| [19] | Fiacchi, M., Gromov hyperbolicity of pseudoconvex finite type domains in \(\mathbb{C}^2\), Math. Ann., 382, 1, 37-68 (2022) ·Zbl 1493.53063 |
| [20] | Kirk, W. A.; Sims, B., Handbook of Metric Fixed Point Theory (2001), Springer Science+Business Media: Springer Science+Business Media Dordrecht ·Zbl 0970.54001 |
| [21] | Hervé, M., Quelques propriétés des applications analytiques d’une boule à m dimensions dan elle-même, J. Math. Pures Appl., 9, 42, 117-147 (1963) ·Zbl 0116.28903 |
| [22] | Julia, G., Mémoire sur l’itération des fonctions rationnelles, J. Math. Pures Appl., 1, 47-245 (1918) ·JFM 46.0520.06 |
| [23] | Karlsson, A., Non-expanding maps and Busemann functions, Ergod. Theory Dyn. Syst., 21, 5, 1447-1457 (2001) ·Zbl 1072.37028 |
| [24] | Karlsson, A.; Noskov, G., The Hilbert metric and Gromov hyperbolicity, Enseign. Math., 48, 1-2, 73-89 (2002) ·Zbl 1046.53026 |
| [25] | Lemmens, B.; Nussbaum, R., Nonlinear Perron-Frobenius Theory, Cambridge Tracts in Mathematics, vol. 189 (2012), Cambridge University Press ·Zbl 1246.47001 |
| [26] | Ostapyuk, O., Backward iteration in the unit ball, Ill. J. Math., 55, 4, 1569-1602 (2011) ·Zbl 1269.30032 |
| [27] | Poggi-Corradini, P., Canonical conjugations at fixed points other than the Denjoy-Wolff point, Ann. Acad. Sci. Fenn. Math., 25, 2, 487-499 (2000) ·Zbl 0958.30012 |
| [28] | Poggi-Corradini, P., Backward iteration sequences with bounded hyperbolic steps for analytic self-maps of the disk, Rev. Mat. Iberoam., 19, 943-970 (2003) ·Zbl 1057.30023 |
| [29] | Rodríguez, J. M.; Tourís, E., A new characterization of Gromov hyperbolicity for negatively curved surfaces, Publ. Mat., 50, 2, 249-278 (2006) ·Zbl 1111.53033 |
| [30] | Webster, C.; Winchester, A., Boundaries of hyperbolic metric spaces, Pac. J. Math., 221, 1, 147-158 (2005) ·Zbl 1177.53042 |
| [31] | Wolff, J., Sur une généralisation d’un thèoréme de Schwarz, C. R. Math. Acad. Sci. Paris, 182, 918-920 (1926) ·JFM 52.0309.05 |
| [32] | Zimmer, A., Gromov hyperbolicity and the Kobayashi metric on convex domains of finite type, Math. Ann., 365, 3-4, 1425-1498 (2016) ·Zbl 1379.53053 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
