Abstract. In this paper we obtain theorems which give the Hausdorff dimension of the invariant set for a finite family of contraction mappings which are "infinitesimal similitudes" on a complete, perfect metric space. Our work generalizes the graph-directed construction of Mauldin and Williams (1988) and is related in its general setting to results of Schief (1996), but differs crucially in that the mappings need not be similitudes. We use the theory of positive linear operators and generalizations of the Krein-Rutman theorem to characterize the Hausdorff dimension as the unique value of σ > 0 for which r(L σ ) = 1, where L σ , σ > 0, is a naturally associated family of positive linear operators and r(L σ ) denotes the spectral radius of L σ . We also indicate how these results can be generalized to countable families of infinitesimal similitudes. The intent here is foundational: to derive a basic formula in its proper generality and to emphasize the utility of the theory of positive linear operators in this setting. Later work will explore the usefulness of the basic theorem and its functional analytic setting in studying questions about Hausdorff dimension.

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In this paper, we study the following boundary value problem involving the weak [Formula: see text]-Laplacian. [Formula: see text] [Formula: see text] where [Formula: see text] is the Sierpiński gasket in [Formula: see text], [Formula: see text] is its boundary, [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] are bounded nonnegative functions. We will show the existence of at least two nontrivial weak solutions to the above problem for a certain range of [Formula: see text] using the analysis of fibering maps on suitable subsets.

In this paper we discuss the continuity of the Hausdorff dimension of the invariant set of generalised graph-directed systems given by contractive infinitesimal similitudes on bounded complete metric spaces. We use the theory of positive linear operators to show that the Hausdorff dimension varies continuously with the functions defining the generalised graph-directed system under suitable assumptions.