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On the rate of convergence of hyperbolic semigroups of holomorphic functions
Author(s) -
Betsakos Dimitrios
Publication year - 2015
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdv022
Subject(s) - mathematics , semigroup , holomorphic function , boundary (topology) , pure mathematics , cancellative semigroup , domain (mathematical analysis) , monotonic function , unit disk , mathematical analysis
Let{ ϕ t } t ⩾ 0be a semigroup of holomorphic self‐maps of the unit disk. We assume that the Denjoy–Wolff point of the semigroup is the point 1; so 1 is the unique attractive boundary fixed point of the semigroup. We further assume that for all t > 0 ,ϕ t ' ( 1 ) 1 (angular derivative), namely the semigroup is hyperbolic. We prove then that the rate of convergence of the semigroup to the point 1, as t → + ∞ , is exponential with exponent arbitrarily close to - π t / ν ( Ω ) , where ν ( Ω ) is the width of the smallest strip containing the associated planar domain Ω . We also prove that the trajectories of the semigroup approach the point 1 in a monotonic way.