Vol. 27, No. 1, 1968

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Geometric theory of a single Markov operator

Robert C. Sine

Vol. 27 (1968), No. 1, 155–166

A Markov operator acting on the space of continuous functions on a compact Hausdorff space which is uniformly stable in the mean allows a topological ergodic decomposition. A partial converse to this is obtained; if the operator has a decomposition it is then uniformly stable in the mean when restricted to the conservative set. The characterization of uniformly mean stable operators in terms of its invariant structures is the major result. The problem of characterizing the manifolds which can be the invariant manifold for some Markov operator is also considered.

Mathematical Subject Classification
Primary: 28.70
Received: 11 August 1967
Published: 1 October 1968
Robert C. Sine