Speeds of convergence of orbits of non-elliptic semigroups of holomorphic self-maps of the unit disk

Filippo Bracci

Abstract


We introduce three quantities related to orbits of non-elliptic continuous semigroups of holomorphic self-maps of the unit disk, the total speed, the orthogonal speed, and the tangential speed and show how they are related and what can be inferred from those.

Keywords


Semigroups of holomorphic functions; hyperbolic geometry; dynamical systems

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References


Abate, M., Iteration Theory of Holomorphic Maps on Taut Manifolds, Mediterranean Press, Rende, 1989.

Aharonov, D., Elin, M., Reich, S., Shoikhet, D., Parametric representation of semicomplete vector fields on the unit balls of Cn and in Hilbert space, Atti Accad. Naz. Lincei 10 (1999), 229–253.

Arosio, L., Bracci, F., Canonical models for holomorphic iteration, Trans. Amer. Math. Soc. 368 (5) (2016), 3305–3339.

Berkson, E., Porta, H., Semigroups of holomorphic functions and composition operators, Michigan Math. J. 25 (1978), 101–115.

Betsakos, D., On the rate of convergence of parabolic semigroups of holomorphic functions, Anal. Math. Phys. 5 (2015), 207–216.

Betsakos, D., Contreras, M. D., Diaz-Madrigal, S., On the rate of convergence of semigroups of holomorphic functions at the Denjoy–Wolff point, to appear in Rev. Mat. Iberoamericana.

Bracci, F., Gumenyuk, P., Contact points and fractional singularities for semigroups of holomorphic self-maps in the unit disc, J. Anal. Math. 130 (1) (2016), 185–217.

Bracci, F., Contreras, M. D., Diaz-Madrigal, S., Topological invariants for semigroups of holomorphic self-maps of the unit disc, J. Math. Pures Appl. 107 (1) (2017), 78–99.

Bracci, F., Contreras, M. D., Diaz-Madrigal, S., On the Koenigs function of semigroups of holomorphic self-maps of the unit disc, Anal. Math. Phys. 8 (4) (2018), 521–540.

Bracci, F., Contreras, M. D., Diaz-Madrigal, S., Gaussier, H., Backward orbits and petals of semigroups of holomorphic self-maps of the unit disc,. Ann. Math. Pura Appl. 198 (2) (2019), 411–441.

Bracci, F., Contreras, M. D., Diaz-Madrigal, S., Gaussier, H., Non-tangential limits and the slope of trajectories of holomorphic semigroups of the unit disc, Trans. Amer. Math. Soc. 373 (2) (2020), 939–969.

Bracci, F., Contreras, M. D., Diaz-Madrigal, S., Gaussier, H., Zimmer, A., Asymptotic behavior of orbits of holomorphic semigroups, J. Math. Pures Appl. doi:10.1016/j.matpur.2019.05.005 online print.

Collingwood, E. F., Lohwater, A. J., The theory of Cluster Sets, Cambridge Tracts in Mathematics and Mathematical Physics, No. 56 Cambridge Univ. Press, Cambridge, 1966.

Contreras, M. D., Diaz-Madrigal, S., Analytic flows on the unit disk: angular derivatives and boundary fixed points, Pacific J. Math., 222 (2005), 253–286.

Contreras, M. D., Diaz-Madrigal, S., Pommerenke, Ch., Fixed points and boundary behavior of the Koenigs function, Ann. Acad. Sci. Fenn. Math. 29 (2004), 471–488.

Contreras, M. D., Diaz-Madrigal, S., Pommerenke, Ch., On boundary critical points for semigroups of analytic functions, Math. Scand. 98 (2006), 125–142.

Cowen, C. C., Iteration and the solution of functional equations for functions analytic in the unit disk, Trans. Amer. Math. Soc. 265 (1981), 69–95.

Elin, M., Jacobzon, F., Parabolic type semigroups: asymptotics and order of contact, Anal. Math. Phys. 4 (2014), 157–185.

Jacobzon, F., Reich, S., Shoikhet, D., Linear fractional mappings, invariant sets, semigroups and commutativity, J. Fixed Point Theory Appl. 5 (2009), 63–91.

Jacobzon, F., Levenshtein, M., Reich, S., Convergence characteristics of oneparameter continuous semigroups, Anal. Math. Phys. 1 (2011), 311–335.

Elin, M., Khavinson, D., Reich, S., Shoikhet, D., Linearization models for parabolic dynamical systems via Abel’s functional equation, Ann. Acad. Sci. Fen. Math. 35 (2010), 439–472.

Elin, M., Levenshtein, M., Reich, S., Shoikhet, D., Commuting semigroups of holomorphic mappings, Math. Scand. 103 (2008), 295–319.

Elin, M., Shoikhet, D., Linearization Models for Complex Dynamical Systems. Topics in Univalent Functions, Functional Equations and Semigroup Theory, Birkhauser, Basel, 2010.

Elin, M., Reich, S., Shoikhet, D., Yacobzon, F., Rates of convergence of oneparameter semigroups with boundary Denjoy–Wolff fixed points, in: Fixed Points Theory and its Applications, Yokohama Publishers, Yokohama, 2008, 43–58.

Elin, M., Shoikhet, D., Zalcman, L., A flower structure of backward flow invariant domains for semigroups, Ann. Acad. Sci. Fenn. Math. 33 (2008), 3–34.

Shoikhet, D., Semigroups in Geometrical Function Theory, Kluwer Academic Publishers, Dordrecht, 2001.

Siskakis, A. G., Semigroups of Composition Operators and the Ces`aro Operator on Hp(D), Ph. D. Thesis, University of Illinois, 1985.

Siskakis, A. G., Semigroups of composition operators on spaces of analytic functions, a review, Contemp. Math. 213, Amer. Math. Soc., Providence, RI, 1998, 229–252.




DOI: http://dx.doi.org/10.17951/a.2019.73.2.21-43
Date of publication: 2020-01-16 07:29:31
Date of submission: 2019-12-31 22:26:40


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