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Subadditive Pressure for IFS with Triangular Maps

100%
Bárány B.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2009
|
tom 57
|
nr 3
263-278
EN
We investigate properties of the zero of the subadditive pressure which is a most important tool to estimate the Hausdorff dimension of the attractor of a non-conformal iterated function system (IFS). Our result is a generalization of the main results of Miao and Falconer [Fractals 15 (2007)] and Manning and Simon [Nonlinearity 20 (2007)].
2
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On the Hausdorff dimension of a family of self-similar sets with complicated overlaps

100%
Bárány B.
Fundamenta Mathematicae
|
2009
|
tom 206
|
nr 1
49-59
EN
We investigate the properties of the Hausdorff dimension of the attractor of the iterated function system (IFS) {γx,λx,λx+1}. Since two maps have the same fixed point, there are very complicated overlaps, and it is not possible to directly apply known techniques. We give a formula for the Hausdorff dimension of the attractor for Lebesgue almost all parameters (γ,λ), γ < λ. This result only holds for almost all parameters: we find a dense set of parameters (γ,λ) for which the Hausdorff dimension of the attractor is strictly smaller.
3
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The absolute continuity of the invariant measure of random iterated function systems with overlaps

63%
Bárány B., Persson T.
Fundamenta Mathematicae
|
2010
|
tom 210
|
nr 1
47-62
EN
We consider iterated function systems on the interval with random perturbation. Let $Y_ε$ be uniformly distributed in [1-ε,1+ ε] and let $f_i ∈ C^{1+α}$ be contractions with fixpoints $a_i$. We consider the iterated function system ${Y_{ε}f_{i} + a_{i}(1-Y_{ε})}ⁿ_{i=1}$, where each of the maps is chosen with probability $p_i$. It is shown that the invariant density is in L² and its L² norm does not grow faster than 1/√ε as ε vanishes. The proof relies on defining a piecewise hyperbolic dynamical system on the cube with an SRB-measure whose projection is the density of the iterated function system.
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