We show that a dissipative, ergodic measure preserving transformation of a σ-finite, non-atomic measure space always has many non-proportional, absolutely continuous, invariant measures and is ergodic with respect to each one of these. §0 Introduction Let (X,B, m, T ) be an invertible, ergodic measure preserving transformation of a σ-finite measure space, then there are no other σ-finite, m-absolutely continuous, T -invariant measure other than constant multiples of m, because the density of any such measure is T -invariant, whence constant by ergodicity. When T is not invertible, the situation becomes more complicated. If (X,B, m, T ) is a conservative, ergodic, measure preserving transformation of a σ-finite measure space, then (again) there are no other σ-finite, m-absolutely continuous, T -invariant measure other than constant multiples of m (see e.g. theorem 1.5.6 in [A]). When T is not conservative, the situation is different. In this note, we show (proposition 1) that a dissipative measure preserving transformation has many non-proportional, σ-finite, absolutely continuous, invariant measures. If the dissipative measure preserving transformation is ergodic (exact), then it is also ergodic (exact) with respect to each of these σ-finite, absolutely continuous, invariant measures (proposition 2). Proposition 1 was known for certain examples: the “Engel series transformation” (see[T], also [S1]); the one sided shift of a random walk on a polycyclic group with centered, adapted jump distribution (ergodicity follows from [K], existence of nonproportional invariant densities follows from [B-E]); and the Euclidean algorithm transformation (see [D-N] which inspired this note). More details are given in §2. §1 is devoted to results (statements and proofs) and §2 has examples of ergodic, dissipative measure preserving transformations. To conclude this introduction, we consider 2000 Mathematics Subject Classification. 37A05, 37A40.
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