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Random affine code tree fractals and Falconer–Sloan condition

Published online by Cambridge University Press:  15 December 2014

ESA JÄRVENPÄÄ ,
MAARIT JÄRVENPÄÄ ,
BING LI  and
ÖRJAN STENFLO
ESA JÄRVENPÄÄ
Affiliation:
Department of Mathematical Sciences, PO Box 3000, 90014 University of Oulu, Finland email esa.jarvenpaa@oulu.fi, maarit.jarvenpaa@oulu.fi
MAARIT JÄRVENPÄÄ
Affiliation:
Department of Mathematical Sciences, PO Box 3000, 90014 University of Oulu, Finland email esa.jarvenpaa@oulu.fi, maarit.jarvenpaa@oulu.fi
BING LI
Affiliation:
Department of Mathematical Sciences, PO Box 3000, 90014 University of Oulu, Finland email esa.jarvenpaa@oulu.fi, maarit.jarvenpaa@oulu.fi Department of Mathematics, South China University of Technology, Guangzhou 510641, PR China email libing0826@gmail.com
ÖRJAN STENFLO
Affiliation:
Department of Mathematics, Uppsala University, PO Box 480, 75106 Uppsala, Sweden email stenflo@math.uu.se
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Abstract

We calculate the almost sure dimension for a general class of random affine code tree fractals in $\mathbb{R}^{d}$. The result is based on a probabilistic version of the Falconer–Sloan condition $C(s)$ introduced in Falconer and Sloan [Continuity of subadditive pressure for self-affine sets. Real Anal. Exchange 34 (2009), 413–427]. We verify that, in general, systems having a small number of maps do not satisfy condition $C(s)$. However, there exists a natural number $n$ such that for typical systems the family of all iterates up to level $n$ satisfies condition $C(s)$.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 5 , August 2016 , pp. 1516 - 1533
Copyright
© Cambridge University Press, 2014 

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