We discuss weak disjointness of homomorphisms of minimal transformation groups and use the techniques involved to deepen our knowledge of the equicontinuous structure relation.

The notions of ergodicity, strong mixing and weak mixing are defined and studied for arbitrary sequences of measure-preserving transformations of a probability space. Several results, notably ones connected with mean ergodic theorems, are generalized from the case of the sequence of all powers of a single transformation to this case. The conditions for ergodicity, strong mixing and weak mixing of sequences of affine transformations of compact groups are investigated.

Only finitely many shift equivalence classes of non-negative aperiodic integral matrices may share a given diagonal Jordan form away from zero. The diagonal assumption is necessary.

An ergodic flow is said to have the weak Pinsker property if it admits a decreasing sequence of factors whose entropies tend to zero and each of which has a Bernoulli complement. We show that this property is preserved under taking factors and d-limits. In addition, we show that a flow has the weak Pinsker property whenever one ergodic transformation in the flow has this property.

Let f be a continuous map of degree one of the circle onto itself. We prove that for every number a from the rotation interval of f there exists an invariant closed set A consisting of points with rotation number a and such that f restricted to A preserves the order. This result is analogous to the one in the case of a twist map of an annulus.

For the invariant sets of dynamical systems a new notion of dimension-the so-called dimension with respect to a dynamical system-is introduced. It has some common features with the general topological notion of the dimension, but it also reflects the dynamical properties of the system. In the one-dimensional case it coincides with the Hausdorff dimension. For multi-dimensional hyperbolic sets formulae for the calculation of our dimension are obtained. These results are generalizations of Manning's results obtained by him for the Hausdorff dimension in the two-dimensional case.

A universal unfolding of a discrete dynamical system f0 is a manifold F of dynamical systems such that each system g sufficiently near f0 is topologically conjugate to an element f of F with the conjugacy φ and the element f depending continuously on f0. An infinitesimally universal unfolding of f0 is (roughly speaking) a manifold F transversal to the topological conjugacy class of f0. Using Nash-Moser iteration we show infinitesimally universal unfoldings are universal and (in part II) give a class of examples relating to moduli of stability introduced by Palis and De Melo.