A class of locally isometric, but not necessarily invertible mappings of an interval is considered. We show that under some conditions the study of the dynamical properties of these mappings can be reduced to interval exchange transformations. On the other hand, there are examples of mappings in this class with ergodic invariant measures supported by Cantor sets. The so-called μβ -sets studied by Y. Katznelson appear naturally in such examples.

For any unitary Zrf-action on a Hilbert space with continuous spectrum weakly wandering vectors are dense. This wandering can be forced to occur along /P-sets. This is a generalization and strengthening of a result due to Krengel. Our method is based on the contractive mapping fixed-point theorem.

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Abstract. E. Hille [Hi1] gave an example of an operator in L 1 [0, 1] satisfying the mean ergodic theorem (MET) and such that sup n T n = ∞ (actually, T n ∼ n 1/4 ). This was the first example of a non-power bounded mean ergodic L 1 operator. In this note, the possible rates of growth (in n) of the norms of T n for such operators are studied. We show that, for every γ > 0, there are positive L 1 operators T satisfying the MET with lim n→∞ T n /n 1−γ = ∞. In the class of positive operators these examples are the best possible in the sense that for every such operator T there exists a γ 0 > 0 such that lim sup n→∞ T n /n 1−γ 0 = 0.A class of numerical sequences {α n }, intimately related to the problem of the growth of norms, is introduced, and it is shown that for every sequence {α n } in this class one can get T n ≥ α n (n = 1, 2, . . .) for some T . Our examples can be realized in a class of positive L 1 operators associated with piecewise linear mappings of [0, 1]. 0. Introduction. The mean ergodic theorem (MET) was originally proved by von Neumann for unitary operators in Hilbert spaces. This theorem triggered a huge number of results, including those extending it to various classes of spaces and operators (see, e.g., [Kr]). We say that a bounded linear operator T in a Banach space X satisfies the MET (or is mean ergodic) if lim n→∞ n −1 n k=1 T k f exists for all f ∈ X. An obvious necessary condition for T to satisfy the MET comes from the classical Banach-Steinhaus theorem; namely, one must have sup n≥1 A n < ∞, where A n = A n (T ) = n −1 n k=1 T k . Such operators T are called Cesàro bounded . A stronger condition, sup n≥1 T n < ∞, which is called power boundedness of T , turns out not to be necessary for the mean ergodicity of T . The first, and nontrivial, example in this direction was given in an old paper of E. Hille [Hi1]. He proved that the operator T defined on0 f (y) dy is mean ergodic, but the norms of the T n grow as n 1/4 . This rate of growth (n 1/4 ) is, of course, related to the concrete analytical nature of Hille's example (more precisely, it is connected with the asymptotics of the Laguerre polynomials, which appear in the kernels of the