Infinitesimal generators of semigroups with prescribed boundary fixed points
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Infinitesimal generators of semigroups with prescribed boundary fixed points Manuel D. Contreras1
· Santiago Díaz-Madrigal1
· Pavel Gumenyuk2
Received: 5 May 2020 / Accepted: 16 July 2020 / Published online: 8 August 2020 © Springer Nature Switzerland AG 2020
Abstract We study infinitesimal generators of one-parameter semigroups in the unit disk D having prescribed boundary regular fixed points. Using an explicit representation of such infinitesimal generators in combination with Krein–Milman Theory we obtain new sharp inequalities relating spectral values at the fixed points with other important quantities having dynamical meaning. We also give a new proof of the classical Cowen–Pommerenke inequalities for univalent self-maps of D. Keywords One-parameter semigroup · Fixed point · Boundary regular fixed point · Infinitesimal generator · Critical point · Spectral value · Value region · Extreme point · Krein–Milman theorem · Cowen–Pommerenke inequalities Mathematics Subject Classification Primary 37C10 · 30C35; Secondary 30D05 · 30C80 · 37F99 · 37C25
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Partially supported by the Ministerio de Economía y Competitividad and the European Union (FEDER) PGC2018-094215-B-100 and by La Junta de Andalucía, FQM-133.
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Santiago Díaz-Madrigal [email protected] Manuel D. Contreras [email protected] Pavel Gumenyuk [email protected]
1
Departamento de Matemática Aplicada II and IMUS, Universidad de Sevilla, Camino de los Descubrimientos, s/n, Sevilla 41092, Spain
2
Department of Mathematics, Milano Politecnico, Via E. Bonardi 9, Milan 20133, Italy
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2.1 Discrete iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 One-parameter semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Herglotz functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Extreme points and Krein–Milman theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Infinitesimal generators of one-parameter semigroups with given boundary regular fixed points 6.1 Representation formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Elliptic semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Non-elliptic semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Extreme points of Genτ (F, ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Loewner–Kufarev-type ODE for self-maps with BRFPs . . . . . . . . . . . . . . . . . . . . . 7.1 Parametric representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Inequalities of Cowen and Pommerenke .
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