Illinois Journal of Mathematics

On circle endomorphisms

R. Cowen

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In [6], Shub and Sullivan posed the problem of finding complete measure theoretic invariants for analytic Lebesgue measure preserving expanding endomorphisms of $S^{1}$. In [2], the author gave necessary and sufficient conditions for two such endomorphisms to be isomorphic. These complete invariants were a mixture of a topological and measure-theoretic nature. Establishing them required finding a coboundary equation with no obvious method of construction. In this note we use a result by Arteaga to furnish a different set of complete isomorphism invariants, still of a mixed topological and measure-theoretic nature but far more easily checked than the ones established in [2].

Article information

Illinois J. Math., Volume 44, Issue 3 (2000), 516-519.

First available in Project Euclid: 20 October 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37E10: Maps of the circle
Secondary: 37A35: Entropy and other invariants, isomorphism, classification 37C15: Topological and differentiable equivalence, conjugacy, invariants, moduli, classification 37F15: Expanding maps; hyperbolicity; structural stability


Cowen, R. On circle endomorphisms. Illinois J. Math. 44 (2000), no. 3, 516--519. doi:10.1215/ijm/1256060411.

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