Amenable relations for endomorphisms

Hawkins, J. M.

doi:10.1090/S0002-9947-1994-1179396-3

Amenable relations for endomorphisms
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by J. M. Hawkins

Trans. Amer. Math. Soc. 343 (1994), 169-191

DOI: https://doi.org/10.1090/S0002-9947-1994-1179396-3

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Abstract:

We give necessary and sufficient conditions for an endomorphism to admit an equivalent invariant $\sigma$-finite measure in terms of a generalized Perron-Frobenius operator. The assumptions are that the endomorphism is nonsingular (preserves sets of measure zero), conservative, and finite-to-1. We study two orbit equivalence relations associated to an endomorphism, and their connections to nonsingularity, ergodicity, and exactness. We also discuss Radon-Nikodym derivative cocycles for the relations and the endomorphism, and relate these to the Jacobian of the endomorphism.

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