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Register a new userOptimal transportation and stationary measures for Iterated Function Systems
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Authors
Beno^it Kloeckner
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In this article we show how ideas, methods and results from optimal transportation can be used to study various aspects of the stationary measuresof Iterated Function Systems equipped with a probability distribution. We recover a classical existence and uniqueness result under a contraction-on-average assumption, prove generalized moment bounds from which tail estimates can be deduced, consider the convergence of the empirical measure of an associated Markov chain, and prove in many cases the Lipschitz continuity of the stationary measure when the system is perturbed, with as a consequence a linear response formula at almost every parameter of the perturbation.
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We study the moments of equilibrium measures for iterated function systems (IFSs) and draw connections to operator theory. Our main object of study is the infinite matrix which encodes all the moment data of a Borel measure on R^d or C. To encode the salient features of a given IFS into precise moment data, we establish an interdependence between IFS equilibrium measures, the encoding of the sequence of moments of these measures into operators, and a new correspondence between the IFS moments and this family of operators in Hilbert space. For a given IFS, our aim is to establish a functorial correspondence in such a way that the geometric transformations of the IFS turn into transformations of moment matrices, or rather transformations of the operators that are associated with them. We first examine the classical existence problem for moments, culminating in a new proof of the existence of a Borel measure on R or C with a specified list of moments. Next, we consider moment problems associated with affine and non-affine IFSs. Our main goal is to determine conditions under which an intertwining relation is satisfied by the moment matrix of an equilibrium measure of an IFS. Finally, using the famous Hilbert matrix as our prototypical example, we study boundedness and spectral properties of moment matrices viewed as Kato-Friedrichs operators on weighted l^2 spaces.
We study the attractor of Iterated Function Systems composed of infinitely many affine, homogeneous maps. In the special case of second generation IFS, defined herein, we conjecture that the attractor consists of a finite number of non-overlapping intervals. Numerical techniques are described to test this conjecture, and a partial rigorous result in this direction is proven.
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