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Qinghe Yin

Australian National University

β-transformations, iterated function systems, and Cantor-type sets

β-transformations and β-expansions were first introduced by Renyi (1957) and further explored by Parry (1961). Given β > 1, the β-transformation Tβ : [0,1) [0,1) is defined by Tβx = βx (mod 1). It possesses an absolutely continuous invariant measure and is ergodic. The β-expansion of x ∈ [0,1) is determined by Tβ. This introduces a symbolic dynamical system Σβ with one-sided shift map. An iterated function system (IFS) is an (n + 1)-tuple (X;f0,,fn-1), where X is a compact metric space and each fi is a contractive map. The theory of IFS’s was first explored by Hutchinson (1981). Many important fractals such as the Sierpinski triangle and the Cantor middle-third set appear as attractors of certain IFS’s. We define the β-attractor Eβ for an IFS as a compact subset, determined by Σβ, of the attractor of the IFS. In the case that all fi are similitudes with contractive ratio 0 < r < 1 and with a disjoint condition, the Hausdorff dimension of Eβ is given by
dim (Eβ) = logβ-.
          - logr

As an application of this result, we compute the Hausdorff dimension for Cantor-type sets constructed from β-expansions with β > 2. This is a joint work with John Hutchinson.