On the Fine Structure of Stationary Measures in Systems Which Contract-on-average
نویسندگان
چکیده
Suppose {f1, . . . , fm} is a set of Lipschitz maps of R . We form the iterated function system (IFS) by independently choosing the maps so that the map fi is chosen with probability pi ( ∑ m i=1 pi = 1). We assume that the IFS contracts on average. We give an upper bound for the Hausdorff dimension of the invariant measure induced on R and as a corollary show that the measure will be singular if the modulus of the entropy ∑ i pi log pi is less than d times the modulus of the Lyapunov exponent of the system. Using a version of Shannon’s Theorem for random walks on semigroups we improve this estimate and show that it is actually attainable for certain cases of affine mappings of R.
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