Identity principles for commuting holomorphic self-maps of the unit disc

Bracci, Filippo; Tauraso, Roberto; Vlacci, Fabio

doi:10.1016/s0022-247x(02)00080-x

Cited by 19 publications

(18 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The same conclusion holds for an arbitrary holomorphic function G on Δ if F commutes with G and satisfies the conditions F (τ )= G(τ )= τ and F (τ ) = G (τ ) = 0 (see, for instance, [10] and [6]). …”

Section: General Rigidity Resultsmentioning

confidence: 60%

See 1 more Smart Citation

Asymptotic Behavior of One-Parameter Semigroups and Rigidity of Holomorphic Generators

Elin

Reich

Shoikhet

et al. 2007

Complex anal.oper. theory

View full text Add to dashboard Cite

We study the asymptotic behavior of one-parameter continuous semigroups of holomorphic mappings. We present angular characteristics of their trajectories at their Denjoy-Wolff points, as well as at their regular repelling points (whenever they exist). This enables us to establish new rigidity properties of holomorphic generators via the asymptotic behavior of the semigroups they generate. Mathematics Subject Classification (2000). Primary 30C45; Secondary 47H20.Let D be a domain (open, connected subset) in the complex plane C, and let Hol (D, C) denote the set of all holomorphic functions (mappings) from D into C.• A family S = {F t } t≥0 of self-mappings of D is called a one-parameter continuous semigroup on D if (i) F t+s (z) = F t (F s (z)) for all t, s ∈ [0, ∞) and z ∈ D;(ii) lim t→0 + F t (z) = z for all z ∈ D. It is well known (see, for example, [4] and [1]) that the continuity condition (ii) implies, in fact, the continuity of S with respect to the parameter at each t ≥ 0. Moreover, by [17], it is also differentiable in [0, ∞) and the limitdefines a holomorphic mapping on D (see also [4,20] and [22]).• The function f defined by (1.1) is called the (infinitesimal) generator of S.

show abstract

Section: General Rigidity Resultsmentioning

confidence: 60%

“…Some generalizations of this result can be found in [3,6,25] and [15]. Of course, if τ ∈ Δ, then f (τ ) exists automatically for f ∈ Hol(Δ, C).…”

Section: General Rigidity Resultsmentioning

confidence: 79%

Asymptotic Behavior of One-Parameter Semigroups and Rigidity of Holomorphic Generators

Elin

Reich

Shoikhet

et al. 2007

Complex anal.oper. theory

View full text Add to dashboard Cite

show abstract

“…This assertion also holds when the unrestricted limit is replaced with the angular one (see [22] and [5]). Recall that a function f ∈ Hol(∆, C) has an angular limit…”

Section: E Of H and Put Hol(d) := Hol(d D)mentioning

confidence: 71%

A rigidity theorem for holomorphic generators on the Hilbert ball

Elin

Levenshtein

Reich

et al. 2008

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. We present a rigidity property of holomorphic generators on the open unit ball B of a Hilbert space H. Namely, if f ∈ Hol(B, H) is the generator of a one-parameter continuous semigroup {F t } t≥0 on B such that for some boundary point τ ∈ ∂B, the admissible limitLet H be a complex Hilbert space with inner product ·, · and induced norm · . If H is finite dimensional, we will identify H with C n . We denote by Hol(D, E) the set of all holomorphic mappings on a domain D ⊂ H which map D into a subset E of H, and put Hol(D) := Hol(D, D).We are concerned with the problem of finding conditions for a mapping F ∈ Hol(D, E) to coincide identically with a given holomorphic mapping on D when they behave similarly in a neighborhood of a boundary point τ ∈ ∂D.A number of basic results in this direction are due to D. M. Burns and S. G. Krantz [6]. They establish conditions at a boundary point for a holomorphic self-mapping F of the open unit disk ∆ := {z ∈ C : |z| < 1} to coincide with the identity mapping (see Proposition 1 below). Then they generalize this fact to the n-dimensional case: for holomorphic self-mappings of the open unit ball (see Proposition 3 below) and of strongly pseudoconvex domains in C n . Further developments of this theme are presented by X. J. Huang in [15], where he obtains similar results for weakly pseudoconvex domains. More recently, L. Baracco, D. Zaitsev and G. Zampieri [3] have proved local boundary rigidity theorems for mappings defined only on one side as germs at a boundary point, and extended their results from boundaries of domains to submanifolds of higher codimension. More higher-dimensional results can be found, for instance, in [2] and [11].In this paper we present a rigidity theorem for holomorphic generators on the open unit ball B of a Hilbert space H which generalizes the analogous theorem for the one-dimensional case [8,17,7] and properly contains the above-mentioned Burns-Krantz theorem for the open unit ball in C n .

show abstract

“…Following [6] and [8], suppose that f ∈ Hol(∆, ∆) and 1 is its Wolff point we say that f ∈ C k (1) if the j-th derivative f (j) extends continuously to ∆ ∪ {1} for all j = 1, . .…”

Section: Definition 22mentioning

confidence: 99%

Boundary constructions of petals at the Wolff point in the parabolic case

Bisi

Gentili

2008

J Anal Math

View full text Add to dashboard Cite

In the same spirit of the classical Leau-Fatou flower theorem, we prove the existence of a petal, with vertex at the Wolff point, for a holomorphic self-map f of the open unit disc ∆ ⊂ C of parabolic type. The result is obtained in the framework of two interesting dynamical situations which require different kinds of regularity of f at the Wolff point τ : f of non-automorphism type and ℜe(f ′′ (τ )) > 0 or f injective of automorphism type, f ∈ C 3+ǫ (τ ) andℜe(f ′′ (τ )) = 0.

show abstract

Identity principles for commuting holomorphic self-maps of the unit disc

Cited by 19 publications

References 15 publications

Asymptotic Behavior of One-Parameter Semigroups and Rigidity of Holomorphic Generators

Asymptotic Behavior of One-Parameter Semigroups and Rigidity of Holomorphic Generators

A rigidity theorem for holomorphic generators on the Hilbert ball

Boundary constructions of petals at the Wolff point in the parabolic case

Contact Info

Product

Resources

About