Pointwise Dimension of Attractors of Iterated Function Systems with Overlap

Brian R. Hunt


In $n$-dimensional Euclidean space, consider iterated function systems that consist of a finite number of $C^{1+\alpha}$ invertible contracting maps $T_i$, and associated probabilities. Such an iterated function system corresponds to a ``baker's map'' in dimension $n+1$, which has a natural invariant measure $\mu$ that attracts Lebesgue almost every initial condition. This measure has an almost everywhere constant pointwise dimension, which as a result coincides with the Hausdorff and information dimensions of the measure. We do not assume that the images of the support of this measure under different $T_i$ are disjoint. The main result of this paper is that if the system is linear or conformal and depends on a number of parameters that include arbitrary translations of the images of the contracting maps, and each map contracts lengths by a factor less than $1/2$, then for Lebesgue almost every set of parameters, the pointwise dimension of $\mu$ is equal to the Lyapunov dimension of $\mu$ (which is an explicit function of the Lyapunov exponents of $\mu$). We also discuss the case of a system that is both nonlinear and nonconformal, and formulate more general ``transversality'' conditions on the parameter dependence that imply the same result.

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Updated: June 30, 1999