APPROXIMATION BY ABSOLUTELY CONTINUOUS INVARIANT MEASURES OF ITERATED FUNCTION SYSTEMS WITH PLACE-DEPENDENT PROBABILITIES

作者：Islam Md Shafiqul^*; Chandler Stephen

来源：Fractals-Complex Geometry Patterns and Scaling in Nature and Society, 2015, 23(4): 1550038.

DOI：10.1142/S0218348X15500383

• Fractals
• IFS
• Random Maps
• The Frobenius - Perron Operator
• Guisti's Bounded Variation
• Absolutely Continuous Invariant Measures

Let S be the attractor (fractal) of a contractive iterated function system (IFS) with place-dependent probabilities. An IFS with place-dependent probabilities is a random map
T = {tau(1)(x), tau(2)(x),..., tau(K)(x); p(1)(x), p(2)(x),..., p(K)(x)},
where the probabilities p(1)(x), p(2)(x),..., p(K)(x) of switching from one transformation to another are functions of positions, that is, at each step, the random map T moves the point x to tau(k)(x) with probability p(k)(x). If the random map T has a unique invariant measure mu, then the support of mu is the attractor S. For a bounded region X subset of R-N, we prove the existence of a sequence {T-0,T-n*} of IFSs with place-dependent probabilities whose invariant measures {mu(n)} are absolutely continuous with respect to Lebesgue measure. Moreover, if X is a compact metric space, we prove that mu(n) converges weakly to mu as n -> infinity. We present examples with computations.

• 出版日期2015-12