Erin P. J. Pearse

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3. (a) If F is as in Theorem 10.7, put A = F′(0), F1(x) = A −1F(x). Then F1(0) = I. Show that F1(x) = Gn◦Gn−1◦. . .◦G1(x) in some neighbourhood of 0, for certain primitive mappings G1, . . . ,Gn. This gives another version of Theorem 10.7: F(x) = F(0)Gn◦Gn−1◦. . .◦G1(x). (b) Prove that the mapping (x, y) 7→ (y, x) of R onto R is not the composition of any(More)
A formula for the interior ε-neighbourhood of the classical von Koch snowflake curve is computed in detail. This function of ε is shown to match quite closely with earlier predictions from [La-vF1] of what it should be, but is also much more precise. The resulting ‘tube formula’ is expressed in terms of the Fourier coefficients of a suitable nonlinear and(More)
An iterated function system Φ consisting of contractive affine mappings has a unique attractor F ⊆ R which is invariant under the action of the system, as was shown by Hutchinson [Hut]. This paper shows how the action of the function system naturally produces a tiling T of the convex hull of the attractor. These tiles form a collection of sets whose(More)
An electrical resistance network (ERN) is a connected graph (G, c). The conductance function cxy weights the edges, which are then interpreted as resistors of possibly varying strengths. The relationship between the natural Dirichlet form E and the discrete Laplace operator ∆ on a finite network is given by E(u, v) = 〈u,∆v〉2 , where the latter is the usual(More)
In a previous paper [21], the authors obtained tube formulas for certain fractals under rather general conditions. Based on these formulas, we give here a characterization of Minkowski measurability of a certain class of self-similar tilings and self-similar sets. Under appropriate hypotheses, self-similar tilings with simple generators (more precisely,(More)
We study the boundary theory of a connected weighted graph G from the viewpoint of stochastic integration. For the Hilbert space HE of Dirichlet-finite functions on G, we construct a Gel’fand triple S ⊆ HE ⊆ S′. This yields a probability measure P on S′ and an isometric embedding of HE into L2(S′,P), and hence gives a concrete representation of the boundary(More)