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Packing spectra for Bernoulli measures supported on Bedford-McMullen carpets

100%
Jordan T., Rams M.
Fundamenta Mathematicae
|
2015
|
tom 229
|
nr 2
171-196
EN
We consider the packing spectra for the local dimension of Bernoulli measures supported on Bedford-McMullen carpets. We show that typically the packing dimension of the regular set is smaller than the packing dimension of the attractor. We also consider a specific class of measures for which we are able to calculate the packing spectrum exactly, and we show that the packing spectrum is discontinuous as a function on the space of Bernoulli measures.
2
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The Hausdorff dimension of the projections of self-affine carpets

81%
Ferguson A., Jordan T., Shmerkin P.
Fundamenta Mathematicae
|
2010
|
tom 209
|
nr 3
193-213
EN
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.
3
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Sets of nondifferentiability for conjugacies between expanding interval maps

81%
Jordan T., Kesseböhmer M., Pollicott M., Stratmann B.
Fundamenta Mathematicae
|
2009
|
tom 206
|
nr 1
161-183
EN
We study differentiability of topological conjugacies between expanding piecewise $C^{1+ϵ}$ interval maps. If these conjugacies are not C¹, then their derivative vanishes Lebesgue almost everywhere. We show that in this case the Hausdorff dimension of the set of points for which the derivative of the conjugacy does not exist lies strictly between zero and one. Moreover, by employing the thermodynamic formalism, we show that this Hausdorff dimension can be determined explicitly in terms of the Lyapunov spectrum. These results then give rise to a "rigidity dichotomy" for the type of conjugacies under consideration.
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