Derivatives of singular inner functions.

Cullen, Michael R.

doi:10.1307/mmj/1029000692

Cited by 19 publications

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“…Recall that a Beurling-Carleson set is a closed subset of the unit circle of zero Lebesgue measure whose complement is a union of arcs k I k with |I k | log 1 |I k | < ∞. To obtain a large supply of constructible measures, we use the following result of Cullen [3]: Lemma 1.5. Suppose the support of µ is contained in a Beurling-Carleson set.…”

Section: Strategymentioning

confidence: 99%

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Prescribing inner parts of derivatives of inner functions

Ivrii

2019

JAMA

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Let J be the set of inner functions whose derivative lies in Nevanlinna class. In this note, we show that the natural map F → Inn(F ) : J / Aut(D) → Inn /S 1 is injective but not surjective. More precisely, we show that that the image consists of all inner functions of the form BS µ where B is a Blaschke product and S µ is the singular factor associated to a measure µ whose support is contained in a countable union of Beurling-Carleson sets. Our proof is based on extending the work of D. Kraus and O. Roth on maximal Blaschke products to allow for singular factors. This answers a question raised by K. Dyakonov.

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Section: Strategymentioning

confidence: 99%

“…In [3], M. Cullen showed that this is the case when the support of µ is a Beurling-Carleson set, that is, a closed subset of the unit circle of zero Lebesgue measure whose complement is a union of arcs k I k with |I k | log 1 |I k | < ∞.…”

Section: Subseqences Of Stable Sequencesmentioning

confidence: 99%

Prescribing inner parts of derivatives of inner functions

Ivrii

2019

JAMA

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“…Several authors have studied conditions on inner functions Φ sufficient to imply that Φ ∈ B γ , and thus, Φ ∈ D 2γ [1,2,3,8,18,20].…”

Section: R a Hibschweilermentioning

confidence: 99%

Composition operators on Dirichlet-type spaces

Hibschweiler

2000

Proc. Amer. Math. Soc.

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“…is exact and has a Bernoulli natural extension, Remark. Ahern and Clark [3] show that if the zeros {a n } of a Blaschke product / satisfy £~= 1 (1 -|a n |) r <oo for some 0< r < h h e n / ' e H l ' r and it follows t h a t / e N. Cullen [6] shows that if/ is a singular inner function in 5F then / ' e H p for 0 < p < \ and hence fsN. In either case if/ satisfies conditions (i) and (iii) of Corollary 3.5, the entropy of/ is given by the formula in this corollary.…”

Section: N F G Martinmentioning

confidence: 99%

On ergodic properties of restrictions of inner functions

Martin

1989

Ergod. Th. Dynam. Sys.

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Abstract. We consider inner functions on the unit disk which have a finite number of singularities on the unit circle. The restriction of such functions to the circle are maps onto the circle. We give sufficient conditions that these restrictions are exact endomorphisms whose natural extensions are Bernoulli and that the entropy is given by Rohlin's formula, /»(/) = Jlog |/'| dp. We also give the entropy in closed form if / is in the Nevalinna class N. An example is considered. In the last section we show that if two restrictions are metrically isomorphic, they are diffeomorphic. IntroductionWe consider restrictions to the unit circle of inner functions on the unit disk which have a finite nonempty set of singularities on the circle. In an earlier paper [11] we studied the maps which were continuous on the circle and showed they were exact endomorphisms whose natural extensions are Bernoulli if their derivative on the circle was larger than one in absolute value.In § 2 we use the results of Rychlik [19] to give a sufficient condition that restrictions of inner functions with a finite nonempty set of singularities are exact endomorphisms whose natural extensions are Bernoulli. In § 3 we give sufficient conditions that the entropy of such restrictions is given by Rohlin's formula, h(f) = Jlog|/'| d\x,. If/' is in the Nevalinna class N we evaluate this integral in terms of the critical points of/ in O and the fixed point of/ in O. In § 4 an example is considered, and in § 5 we show that metrically isomorphic restrictions are conformally conjugate to within a rotation.The study of ergodic properties of inner functions was begun by Aaronson [1], [2] and Neuwirth [14], although Adler and Kemperman had studied special inner functions prior to these papers. Subsequent work can be found in

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Derivatives of singular inner functions.

Cited by 19 publications

References 0 publications

Prescribing inner parts of derivatives of inner functions

Prescribing inner parts of derivatives of inner functions

Composition operators on Dirichlet-type spaces

On ergodic properties of restrictions of inner functions

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