Borel automorphisms with no finite invariant measure

NONINVERTIBLE TRANSFORMATIONS ADMITTING NO ABSOLUTELY CONTINUOUS ct-FINITE INVARIANT MEASURE

We study a family of H-to-1 conservative ergodic endomorphisms which we will show to admit no rj-finite absolutely continuous invariant measure. We exhibit recurrent measures for these transformations and study their ratio sets; the examples can be realized as C°° endomorphisms of the 2-torus.

Groups of Automorphisms of Borel Spaces

Identifying Q-processes with a given Finite Μ-invariant Measure

Let Q = (qij , i, j ∈ S) be a stable and conservative Q-matrix over a state space S consisting of an irreducible (transient) class C and a single absorbing state 0, which is accessible fromC. Suppose thatQ admits a finiteμ-subinvariant measure m = (mj , j ∈ C) on C. We consider the problem of identifying all Q-processes for which m is a μ-invariant measure on C.

Non-isomorphism Invariant Borel Quantifiers

Every isomorphism invariant Borel subset of the space of structures on the natural numbers in a countable relational language is definable in Lω1ω by a theorem of Lopez-Escobar. We derive variants of this result for stabilizer subgroups of the symmetric group Sym(N) for families of relations and non-isomorphism invariant generalized quantifiers on the natural numbers such as “for all even numbe...

Non-permutation Invariant Borel Quantifiers

Every permutation invariant Borel subset of the space of countable structures is definable in Lω1ω by a theorem of Lopez-Escobar. We prove variants of this theorem relative to fixed relations and fixed non-permutation invariant quantifiers. Moreover we show that for every closed subgroup G of the symmetric group S∞, there is a closed binary quantifier Q such that the G-invariant subsets of the ...