On the Fine Structure of Stationary Measures in Systems Which Contract-on-Average

Matthew Nicol (), Nikita Sidorov () and David Broomhead ()
Additional contact information
Matthew Nicol: University of Surrey
Nikita Sidorov: UMIST
David Broomhead: UMIST

Journal of Theoretical Probability, 2002, vol. 15, issue 3, 715-730

Abstract: Abstract Suppose {f 1,...,f m } is a set of Lipschitz maps of ℝ d . We form the iterated function system (IFS) by independently choosing the maps so that the map f i is chosen with probability p i (∑ m i=1 p i =1). We assume that the IFS contracts on average. We give an upper bound for the upper Hausdorff dimension of the invariant measure induced on ℝ d and as a corollary show that the measure will be singular if the modulus of the entropy ∑ i p i log p i 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.

Keywords: Iterated function system; stationary measure; Hausdorff dimension; entropy; random walk (search for similar items in EconPapers)
Date: 2002
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Citations: View citations in EconPapers (1)

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DOI: 10.1023/A:1016224000145

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