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On ergodic properties of restrictions of inner functions

Published online by Cambridge University Press:  19 September 2008

N.F.G. Martin
N.F.G. Martin
Affiliation:
Department of Mathematics, Math-Astronomy Building, University of Virginia, Charlottesville, VA 22903, USA
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Abstract

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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, We also give the entropy in closed form if ƒ' 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.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 9 , Issue 1 , March 1989 , pp. 137 - 151
Copyright
Copyright © Cambridge University Press 1989

References

REFERENCES

1.Aaronson, Jon. Ergodic theory of inner functions of the upper half plane. Ann. Inst. Henri Poincairé 14 (1978), 233–253.Google Scholar
2.Aaronson, Jon. A remark on exactness of inner functions. J. London Math. Soc. (2)23 (1981), 469–474.CrossRefGoogle Scholar
3.Ahern, P. R. & Clark, D. N.. On inner functions with H p-derivative. Michigan Math. J. 21 (1974), 115–127.CrossRefGoogle Scholar
4.Burckel, R. B.. Iterating analytic self maps of discs. Amer. Math. Monthly 88 (1981), 387–460.CrossRefGoogle Scholar
5.Caratheodory, C.. Theory of Functions. Vol. II. Chelsea: New York, 1954.Google Scholar
6.Cullen, M. R.. Derivatives of singular inner functions. Michigan Math. J. 18 (1971), 283–287.CrossRefGoogle Scholar
7.Fernàndez, J. L.. A note on entropy and inner functions. Israel J. Math. 53 (1986), 158–162.CrossRefGoogle Scholar
8.Forelli, F.. The isometries of H p. Canad. J. Math. 16 (1964), 721–728.CrossRefGoogle Scholar
9.Garnett, J. B.. Bounded Analytic Functions. Academic Press: New York, 1981.Google Scholar
10.Letac, G.. Which functions preserve Cauchy laws? Proc. AMS 67 (1977), 277–286.CrossRefGoogle Scholar
11.Martin, N. F. G.. On finite Blaschke products whose restrictions to the unit circle are exact endomorphisms. Bull. London Math. Soc. 15 (1983), 343–348.CrossRefGoogle Scholar
12.Martin, N. F. G.. Classification of some finite Blaschke products as metric endomorphisms. J. Math. Anal. Appl. 114 (1986), 205–209.CrossRefGoogle Scholar
13.Martin, N. F. G. and England, J. W.. Mathematical theory of entropy. Encyl. of Math, and Appl. Vol. 12. Addison-Wesley: New York, 1981.Google Scholar
14.Neuwirth, J. H.. Ergodicity of some mappings of the circle and the line. Israel J. Math. 317 (1978), 359–367.CrossRefGoogle Scholar
15.Parry, W.. Entropy and generators in ergodic theory. Math. Lecture Note Series. Benjamin; New York, 1969.Google Scholar
16.Petit, B.. Theorie ergodic: Classification de certaines transformation réeles. Ann. Inst. Henri Poincaire 15 (1979), 25–32.Google Scholar
17.Pommerenke, C.. On ergodic properties of inner functions. Math. Ann. 256 (1981), 43–50.CrossRefGoogle Scholar
18.Rudin, W.. Real and Complex Analysis. 3rd Ed.McGraw-Hill: New York, 1987.Google Scholar
19.Rychlik, M.. Bounded variation and invariant measures. Studia Math. 86 (1983), 69–80.CrossRefGoogle Scholar
20.Shub, M. & Sullivan, D.. Expanding endomorphisms of the circle revisited. Ergod. Th. & Dynam. Sys. 5 (1985), 285–289.CrossRefGoogle Scholar
21.Wolff, J.. Sur une généralisation d'un théorème de Schwartz. C. R. Acad. Sci. Paris 182 (1926), 918–920 and 183 (1926), 500–502.Google Scholar