Absolutely Continuous, Invariant Measures for Dissipative, Ergodic Transformations

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. 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]). In this note, we show (proposition 1) that a dissipative, ergodic measure preserving transformation has many non-proportional, σ-finite, absolutely continuous, invariant measures and is ergodic with respect to each of them (proposition 2). This result was previously known for the one sided shift of a random walk on a polycyclic group with centered, adapted jump distribution (ergodicity is shown in [K], the existence of non-proportional invariant densities follows from [B-E]); and the Euclidean algorithm transformation (see [D-N] which inspired this note). To conclude this introduction, we consider An illustrative example. Fix p ∈ (0, 1) and consider the stochastic matrix p : Z × Z → [0, 1] defined by ps,s := 1 − p, ps,s+1 := p and ps,t = 0 ∀ t 6= s, s + 1. Let (X,B, m, T ) be the one-sided Markov shift with X := Z, B the σ-algebra generated by cylinders (i.e. sets of form [a1, . . . , ak] := {x ∈ X : xj = aj ∀ 1 ≤ j ≤ k} and m : B → [0,∞] the measure satisfying m([a1, . . . , ak]) := ∏k−1 j=1 paj ,aj+1 . It is not hard to check that 2000 Mathematics Subject Classification. 37A05, 37A40.