Hostname: page-component-848d4c4894-x5gtn Total loading time: 0 Render date: 2024-06-08T06:12:55.009Z Has data issue: false hasContentIssue false
  • English
  • Français

The Bernoulli property of inner functions

Published online by Cambridge University Press:  19 September 2008

Marcos Craizer
Marcos Craizer
Affiliation:
Departamento de Matemática, Pontifícia Universidade Católica, Rua Marquês de São Vicente, 225, Rio de Janeiro - R.J., Brasil
Get access Rights & Permissions [Opens in a new window]

Abstract

Let f: D → D be an inner function with a fixed point p ∈ D, and f*: S1 → S1 be its extension to the unit circle. We prove in this paper that the Rohlin invertible extension of the system (f*, λp) is equivalent to a generalized Bernoulli shift, where λp is the harmonic measure associated with p.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 12 , Issue 2 , June 1992 , pp. 209 - 215
Copyright
Copyright © Cambridge University Press 1992

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[A1]Aaronson, J.. Ergodic theory for inner functions of the upper half plane. Ann. Inst. Henri Poincaré, 14 (1978), 233–253.Google Scholar
[A2]Aaronson, J.. A remark on the exactness of inner functions. J. London Math. Soc. 23 (1981), 469–474.CrossRefGoogle Scholar
[C]Craizer, M.. Entropy of inner functions. Israel J. Math. 74 (1991), 129–168.CrossRefGoogle Scholar
[F–O]Friedman, N. & Ornstein, D.. On the isomorphism of weak Bernoulli transformations. Adv. Math. 5 (1970), 365–394.CrossRefGoogle Scholar
[M]Martin, N. F. G.. On ergodic properties of restrictions of inner functions. Ergod. Th. & Dynam. Sys., 9 (1989), 137–151.CrossRefGoogle Scholar
[N]Neuwirth, J. H.. Ergodicity of some mappings of the circle and the line. Israel J. Math. 31 (1978), 359–367.CrossRefGoogle Scholar
[O1]Ornstein, D.. Ergodic Theory, Randomness and Dynamical Systems. (Yale University Press: 1974).Google Scholar
[O2]Ornstein, D.. Two Bernoulli shifts with infinite entropy are isomorphic. Adv. Math. 5 (1971) 339–348.CrossRefGoogle Scholar
[P]Petit, B.. Théorie ergodique: classification de certaines transformations réelles. Ann. Inst. H. Poincaré XV (1979), 25–32.Google Scholar
[Po1]Pommerenke, Ch.. On the iteration of analytic functions in a halfplane, I. J. London Math. Soc. 19 (1979), 439–447.CrossRefGoogle Scholar
[Po2]Pommerenke, Ch.. On ergodic properties of inner functions. Math. Ann. 256 (1981), 43–50.CrossRefGoogle Scholar
[R]Rohlin, V. A.. Exact endomorphisms of a Lebesgue space. Amer. Math. Soc. Trans. 39 (1964), 1–36.Google Scholar