Approximation of measures by Markov processes and homogeneous affine iterated function systems

1989 ◽  
Vol 5 (1) ◽  
pp. 69-87 ◽  
Author(s):  
John H. Elton ◽  
Zheng Yan
Keyword(s):  
Markov Processes ◽  
Iterated Function Systems ◽  
Iterated Function

FRACTALS, SEMIFRACTALS AND MARKOV OPERATORS

1999 ◽  
Vol 09 (02) ◽  
pp. 307-325 ◽  
Author(s):  
A. LASOTA ◽  
J. MYJAK
Keyword(s):  
Markov Processes ◽  
Metric Space ◽  
Iterated Function Systems ◽  
Markov Operators ◽  
Iterated Function

The paper contains a review of results concerning the theory of iterated function systems (IFS) acting on an arbitrary metric space (without any assumption of compactness). First we discuss IFS acting on sets and we define fractals and semifractals using topological limits. Then we study IFS with probabilities acting on measures and we show a relationship with the theory of Markov operators and Markov processes.

scholarly journals Hausdorff measure and Assouad dimension of generic self-conformal IFS on the line

2020 ◽  
pp. 1-31
Author(s):  
Balázs Bárány ◽  
Károly Simon ◽  
István Kolossváry ◽  
Michał Rams
Keyword(s):  
Hausdorff Measure ◽  
Similar Case ◽  
Iterated Function Systems ◽  
The Self ◽  
Dimensional Hausdorff Measure ◽  
Assouad Dimension ◽  
Iterated Function ◽  
Self Similar ◽  
Weak Separation ◽  
Topological Sense

This paper considers self-conformal iterated function systems (IFSs) on the real line whose first level cylinders overlap. In the space of self-conformal IFSs, we show that generically (in topological sense) if the attractor of such a system has Hausdorff dimension less than 1 then it has zero appropriate dimensional Hausdorff measure and its Assouad dimension is equal to 1. Our main contribution is in showing that if the cylinders intersect then the IFS generically does not satisfy the weak separation property and hence, we may apply a recent result of Angelevska, Käenmäki and Troscheit. This phenomenon holds for transversal families (in particular for the translation family) typically, in the self-similar case, in both topological and in measure theoretical sense, and in the more general self-conformal case in the topological sense.

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