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Abstract: We study iterated function systems (IFSs) of contractive similitudes on with overlaps. We introduce a generalized finite type condition which extends a more restrictive condition in [S.-M. Ngai, Y. Wang, Hausdorff dimension of self-similar sets with overlaps, J. London Math. Soc. (2) 63 (3) (2001) 655–672] and allows us to include some IFSs of contractive similitudes whose contraction ratios are not exponentially commensurable. We show that the generalized finite type condition implies the weak separation property. Under this condition, we can identify the attractor of the IFS with that of a graph-directed IFS, and by modifying a setup of Mauldin and Williams [R.D. Mauldin, S.C. Williams, Hausdorff dimension in graph directed constructions, Trans. Amer. Math. Soc. 309 (1988) 811–829], we can compute the Hausdorff dimension of the attractor in terms of the spectral radius of certain weighted incidence matrix. [Copyright &y& Elsevier]

For an expanding integral s×s matrix A with &z.sfnc;detA&z.sfnc;=p, it is well known that if D={d0,…,dp−1}⊂Zs is a complete set of coset representatives of Zs/AZs, then T(A,D) is a self-affine tile. In this paper we show that if p is a prime, such D actually characterizes the tile digit sets provided that span(D)=Rs. This result is known for s=1, the one-dimensional case [R. Kenyon, in: Contemp. Math., vol. 135, 1992, pp. 239–264] and the question for s>1 has been considered by Lagarias and Wang [J. London Math. Soc. 53 (1996) 21–49] under some other conditions. The proof here involves a new setup to study the zeros of the mask m(ξ)=p−1∑j=0p−1e2πi〈ξ,dj〉. It can also be generalized to consider the existence of a compactly supported L1-solution of the refinement equation (scaling function) with positive coefficients. [Copyright &y& Elsevier]

The Cauchy transform of a measure has been used to study the analytic capacity and uniform rectifiability of subsets in C. Recently, Lund et al. (Experiment. Math. 7 (1998) 177) have initiated the study of such transform F of self-similar measure. In this and the forecoming papers (Starlikeness and the Cauchy transform of some self-similar measures, in preparation; The Cauchy transform on the Sierpinski gasket, in preparation), we study the analytic and geometric behavior as well as the fractal behavior of the transform F. The main concentration here is on the Laurent coefficients {an}n=0∞ of F. We give asymptotic formulas for {an}n=0∞ and for F(k)(z) near the support of μ, hence the precise growth rates on |an| and |F(k)| are determined. These formulas are connected with some multiplicative periodic functions, which reflect the self-similarity of μ and K. As a by-product, we also discover new identities of certain infinite products and series. [Copyright &y& Elsevier]

Abstract: We obtain the equivalence conditions for an on-diagonal upper bound of heat kernels on self-similar measure energy spaces. In particular, this upper bound of the heat kernel is equivalent to the discreteness of the spectrum of the generator of the Dirichlet form, and to the global Poincaré inequality. The key ingredient of the proof is to obtain the Nash inequality from the global Poincaré inequality. We give two examples of families of spaces where the global Poincaré inequality is easily derived. They are the post-critically finite (p.c.f.) self-similar sets with harmonic structure and the products of self-similar measure energy spaces. [Copyright &y& Elsevier]