Let X be a compact metric space and H(X) the metric space of continuous selfmaps of X. The subset H(X,m) is then defined as follows: A transformation T belongs to H(X,m) provided there is a T-invariant probability measue mu such that T is mixing for the measure mu.

Example (S.Siboni) For a certain space X, H(m) in't closed.

Theorem 1 H(m) is always an analytic set.

Theorem 2 For a certain space Y, H(m) is a complete analytic

subset of H.

The space Y is immense, but further effort yields an example in which X is a Cantor set and Y has dimenion 1.