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1974•Mathematics of the USSR-IzvestiyaOpen access

QUASI-INVARIANT MEASURES FOR TOPOLOGICAL DYNAMICAL SYSTEMS

Isaac Kornfeld

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Abstract

It is proved that for a topological dynamical system to admit an ergodic quasi-invariant measure of type III (a measure which is not equivalent to any σ-finite invariant measure) it is necessary and sufficient that this system have a recurrent point. For systems with a recurrent point, it is shown that there exist a nondenumerable number of pairwise singular ergodic quasi-invariant measures of type III. Bibliography: 6 items.

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It is proved that for a topological dynamical system to admit an ergodic quasi-invariant measure of type III (a measure which is not equivalent to any σ-finite invariant measure) it is necessary and sufficient that this system have a recurrent point. For systems with a recurrent point, it is shown that there exist a nondenumerable number of pairwise singular ergodic quasi-invariant measures of type III. Bibliography: 6 items.

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Available abstract

It is proved that for a topological dynamical system to admit an ergodic quasi-invariant measure of type III (a measure which is not equivalent to any σ-finite invariant measure) it is necessary and sufficient that this system have a recurrent point. For systems with a recurrent point, it is shown that there exist a nondenumerable number of pairwise singular ergodic quasi-invariant measures of type III. Bibliography: 6 items.

Key concepts: Invariant (physics), Dynamical systems theory, Topological dynamics, Topology (electrical circuits), Mathematics, Pure mathematics, Physics, Mathematical physics

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