Abstract.Two results on site percolation on the d-dimensional lattice, d^l arbitrary, are presented. In the first theorem, we show that for stationary underlying probability measures, each infinite cluster has a well-defined density with probability one. The second theorem states that if in addition, the probability measure satisfies the finite energy condition of Newman and Schulman, then there can be at most one infinite cluster with probability one. The simple arguments extend to a broad class of finite-dimensional models, including bond percolation and regular lattices.Our notation is as follows. Let Z d be the d-dimensional lattice, with d^.1. Finite d-dimensional boxes in Z d whose sides are parallel to the coordinate directions are called rectangles. The number of points in a finite set F is denoted by φ (F). We set and we let μ denote a probability measure on X which is stationary, i.e. invariant under translation by each element of Z d . For each xεX, the connected components of the nearest neighbor graph whose set of vertices is

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Abstract. Let T be an n-to-1 covering transformation of the compact metric space X (e.g. (X, T) the n-shift). For suitable functions g on X an "inverse" r of Tis defined: ,:p,, is a Markov kernel. If g is strictly positive and satisfies a Lipschitz condition, then there exists a unique ~&-invariant measure, strongly mixing under T. Conversely, we associate to any T-invariant probability measure a suitable g, and ifg is "nice", then strong mixing is present. Examples include all Bernoulli and Markov measures on the n-shift. The strong mixing criterion is useful, and applications to harmonic analysis, ergodic theory, and symbolic dynamics are given. For example: if (B is any infinite subgroup of the group of roots of unity, there exist uncountably many (explicitly constructible) continuous Morse sequences whose corresponding dynamical systems are pairwise non-isomorphic and all have as eigenvalue group exactly the given group ~.

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