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The flow of viscous, particle-laden wetting thin films on an inclined plane is studied experimentally as the concentration is increased to the maximum packing limit. The slurry is a non-neutrally buoyant mixture of silicone oil and either solid glass beads or glass bubbles. At low concentrations ͑␾ Ͻ 0.45͒, the elapsed time versus average front position(More)
A general framework for investigating topological actions of Z d on compact metric spaces was proposed by Boyle and Lind in terms of expansive behavior along lower-dimensional subspaces of R d. Here we completely describe this expansive behavior for the class of algebraic Z d-actions given by commuting automorphisms of compact abelian groups. The(More)
Analogues of the prime number theorem and Merten's theorem are well-known for dynamical systems with hyperbolic behaviour. In this paper a 3-adic extension of the circle doubling map is studied. The map has a 3-adic eigendirection in which it behaves like an isometry, and the loss of hyperbolicity leads to weaker asymptotic results on orbit counting than(More)
Let ξ 1 ,. .. , ξ r be complex numbers with K = Q(ξ 1 ,. .. , ξ r) having tran-scendence degree r − 1 over Q. Consider the equation a 1 x 1 + · · · + a k x k = 1, (1) in which the a i 's are fixed elements of K × , no proper subsum a i 1 x i 1 + · · · + a i j x i j vanishes, and we seek solutions x i ∈ Γ = ξ 1 ,. .. , ξ r. It is well–known that (1) has only(More)
We introduce a class of group endomorphisms – those of finite combinatorial rank – exhibiting slow orbit growth. An associated Dirichlet series is used to obtain an exact orbit counting formula, and in the connected case this series is shown to to be a rational function of exponential variables. Analytic properties of the Dirichlet series are related to(More)
Let α be a Z d –action (d ≥ 2) by automorphisms of a compact metric abelian group. For any non–linear shape I ⊂ Z d , there is an α with the property that I is a minimal mixing shape for α. The only implications of the form " I is a mixing shape for α =⇒ J is a mixing shape for α " are trivial ones for which I contains a translate of J. If all shapes are(More)