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We show that one-dimensional semilinear second-order parabolic equations have finite-dimensional dynamics on attractors. In particular, this is true for reaction-diffusion equations with convection on (0, 1). We obtain new topological criteria for a class of dissipative equations of parabolic type in Banach spaces to have finite-dimensional dynamics on invariant compact sets. The dynamics of these equations on an attractor A is finite-dimensional (can be described by an ordinary differential equation) if A can be embedded in a finitedimensional C 1-submanifold of the phase space.

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Abstract. For a continuous semicascade on a metrizable compact set Ω, we consider the weak * convergence of generalized operator ergodic means in End C * (Ω). We discuss conditions on the dynamical system under which (a) every ergodic net contains a convergent subsequence; (b) all ergodic nets converge; (c) all ergodic sequences converge. We study the relationships between the convergence of ergodic means and the properties of transitivity of the proximality relation on Ω, minimality of supports of ergodic measures, and uniqueness of minimal sets in the closure of trajectories of a semicascade.These problems are solved in terms of three algebraic-topological objects associated with the dynamical system: the Ellis enveloping semigroup, the Köhler operator semigroup Γ, and the semigroup G that is the weak * closure of the convex hull of Γ in End C * (Ω). The main results are stated for ordinary semicascades (whose Ellis semigroup is metrizable) and tame semicascades.For a dynamics, being ordinary is equivalent to being "nonchaotic" in an appropriate sense.We present a classification of compact dynamical systems in terms of topological properties of the above-mentioned semigroups.