Displaying similar documents to “On univoque points for self-similar sets”

On the Hausdorff dimension of a family of self-similar sets with complicated overlaps

Balázs Bárány (2009)

Fundamenta Mathematicae

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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...

The dimension of graph directed attractors with overlaps on the line, with an application to a problem in fractal image recognition

Michael Keane, K. Károly, Boris Solomyak (2003)

Fundamenta Mathematicae

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Consider a graph directed iterated function system (GIFS) on the line which consists of similarities. Assuming neither any separation conditions, nor any restrictions on the contractions, we compute the almost sure dimension of the attractor. Then we apply our result to give a partial answer to an open problem in the field of fractal image recognition concerning some self-affine graph directed attractors in space.

On the attractors of Feigenbaum maps

Guifeng Huang, Lidong Wang (2014)

Annales Polonici Mathematici

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A solution of the Feigenbaum functional equation is called a Feigenbaum map. We investigate the likely limit set (i.e. the maximal attractor in the sense of Milnor) of a non-unimodal Feigenbaum map, prove that it is a minimal set that attracts almost all points, and then estimate its Hausdorff dimension. Finally, for every s ∈ (0,1), we construct a non-unimodal Feigenbaum map with a likely limit set whose Hausdorff dimension is s.

Inhomogeneous self-similar sets and box dimensions

Jonathan M. Fraser (2012)

Studia Mathematica

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We investigate the box dimensions of inhomogeneous self-similar sets. Firstly, we extend some results of Olsen and Snigireva by computing the upper box dimensions assuming some mild separation conditions. Secondly, we investigate the more difficult problem of computing the lower box dimension. We give some non-trivial bounds and provide examples to show that lower box dimension behaves much more strangely than upper box dimension, Hausdorff dimension and packing dimension.

Separation conditions on controlled Moran constructions

Antti Käenmäki, Markku Vilppolainen (2008)

Fundamenta Mathematicae

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It is well known that the open set condition and the positivity of the t-dimensional Hausdorff measure are equivalent on self-similar sets, where t is the zero of the topological pressure. We prove an analogous result for a class of Moran constructions and we study different kinds of Moran constructions in this respect.

Infinite Graph-Directed Systems and Hausdorff Dimension

Amit Priyadarshi (2017)

Waves, Wavelets and Fractals

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In this paper we study infinite graph-directed iterated function systems on compact metric spaces given by contractive ‘infinitesimal similitudes’. We derive formula for the Hausdorff dimension of the ‘invariant set’ for such a system in terms of the spectral radii of the naturally associated family of the ‘Perron- Frobenius operators’. The results in this paper generalizes the results obtained in [20], where finite graphdirected systems and infinite iterated function systems are considered ...

Subadditive Pressure for IFS with Triangular Maps

Balázs Bárány (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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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)].

The Hausdorff dimension of the projections of self-affine carpets

Andrew Ferguson, Thomas Jordan, Pablo Shmerkin (2010)

Fundamenta Mathematicae

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We study the orthogonal projections of a large class of self-affine carpets, which contains the carpets of Bedford and McMullen as special cases. Our main result is that if Λ is such a carpet, and certain natural irrationality conditions hold, then every orthogonal projection of Λ in a non-principal direction has Hausdorff dimension min(γ,1), where γ is the Hausdorff dimension of Λ. This generalizes a recent result of Peres and Shmerkin on sums of Cantor sets.

Universal measure zero, large Hausdorff dimension, and nearly Lipschitz maps

Ondřej Zindulka (2012)

Fundamenta Mathematicae

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We prove that each analytic set in ℝⁿ contains a universally null set of the same Hausdorff dimension and that each metric space contains a universally null set of Hausdorff dimension no less than the topological dimension of the space. Similar results also hold for universally meager sets. An essential part of the construction involves an analysis of Lipschitz-like mappings of separable metric spaces onto Cantor cubes and self-similar sets.

Contracting-on-Average Baker Maps

Michał Rams (2006)

Bulletin of the Polish Academy of Sciences. Mathematics

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We estimate from above and below the Hausdorff dimension of SRB measure for contracting-on-average baker maps.

Hausdorff gaps and towers in 𝓟(ω)/Fin

Piotr Borodulin-Nadzieja, David Chodounský (2015)

Fundamenta Mathematicae

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We define and study two classes of uncountable ⊆*-chains: Hausdorff towers and Suslin towers. We discuss their existence in various models of set theory. Some of the results and methods are used to provide examples of indestructible gaps not equivalent to a Hausdorff gap. We also indicate possible ways of developing a structure theory for towers based on classification of their Tukey types.