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The pseudo-isotopy techniques of Cerf-Hatcher-Wagoner are combined with surgery theory to give information about the group of isotopy classes of diffeomorphisms of a smooth manifold. For example, for the w-torus, n ^ 6, this group is determined completely. We also provide a geometric interpretation of the periodicity sequence of [11].Introduction. Let M be an %-dim (n ^ 6) smooth manifold without boundary 1 and let Diff M be the group of diffeomorphisms of M. Let Aut M be the iί-space of simple homotopy equivalences of M to itself, i.e., Aut M-{feM M \f is a simple homotopy equivalence}. We have the following fibration

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1. Introduction. Let T be a conservative positive contraction on the Lx space of a finite measure space (X, &, ft). A theorem of Chacon [5], [2] shows that T defines a sub CT-field J of J^ consisting of invariant subsets of X. The ratio ergodic limits are measurable with respect to J [5], [2] and the class of these limits contains Lm(X, J, ft), which can be considered as the invariant functions of the adjoint transformation [2]. The main purpose of the present paper is to show that any positive contraction on LX(X, 3F, ft) behaves, asymptotically, like a conservative transformation (Theorems 3 and 4) and that the invariant functions of the adjoint transformation can be approximated by the ratio ergodic limits.Intuitively, a ratio ergodic limit corresponds to the result of an averaging process of different values of a function. It is then natural to consider these limits as functions that are smooth with respect to the asymptotic behaviour of the transformation. This leads (Theorem 6) to a Martin-Doob type representation [12], [8] of invariant functions as the L» functions of a compact Hausdorff space JÏ with a Baire measure. The topology on Ji is just strong enough to make the ergodic limits to correspond to continuous functions. As an example we consider a transformation of Feller [10] and show that for this case the above representation is identical with the Poisson representation of harmonic functions in the unit disk. We also consider the possibility of joining X and JÍ, convergence of measures to i' in IuJ (Theorem 7), and a relation (Lemma 9) between the Feller and Martin-Doob type representations, corresponding to a result of Feldman [9].