Parabolic Cantor sets

Mariusz Urbański

Fundamenta Mathematicae (1996)

  • Volume: 151, Issue: 3, page 241-277
  • ISSN: 0016-2736

Abstract

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The notion of a parabolic Cantor set is introduced allowing in the definition of hyperbolic Cantor sets some fixed points to have derivatives of modulus one. Such difference in the assumptions is reflected in geometric properties of these Cantor sets. It turns out that if the Hausdorff dimension of this set is denoted by h, then its h-dimensional Hausdorff measure vanishes but the h-dimensional packing measure is positive and finite. This latter measure can also be dynamically characterized as the only h-conformal measure. It is relatively easy to see that any two parabolic Cantor sets formed with the help of the same alphabet are canonically topologically conjugate and we then discuss the rigidity problem of what are the possibly weakest sufficient conditions for this topological conjugacy to be "smoother". It turns out that if the conjugating homeomorphism preserves the moduli of the derivatives at periodic points, then the dimensions of both sets are equal and the homeomorphism is shown to be absolutely continuous with respect to the corresponding h-dimensional packing measures. This property in turn implies the conjugating homeomorphism to be Lipschitz continuous. Additionally the existence of the scaling function is shown and a version of the rigidity theorem, expressed in terms of scaling functions, is proven. We also study the real-analytic Cantor sets for which the stronger rigidity can be shown, namely that the absolute continuity of the conjugating homeomorphism alone implies its real analyticity.

How to cite

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Urbański, Mariusz. "Parabolic Cantor sets." Fundamenta Mathematicae 151.3 (1996): 241-277. <http://eudml.org/doc/212194>.

@article{Urbański1996,
abstract = {The notion of a parabolic Cantor set is introduced allowing in the definition of hyperbolic Cantor sets some fixed points to have derivatives of modulus one. Such difference in the assumptions is reflected in geometric properties of these Cantor sets. It turns out that if the Hausdorff dimension of this set is denoted by h, then its h-dimensional Hausdorff measure vanishes but the h-dimensional packing measure is positive and finite. This latter measure can also be dynamically characterized as the only h-conformal measure. It is relatively easy to see that any two parabolic Cantor sets formed with the help of the same alphabet are canonically topologically conjugate and we then discuss the rigidity problem of what are the possibly weakest sufficient conditions for this topological conjugacy to be "smoother". It turns out that if the conjugating homeomorphism preserves the moduli of the derivatives at periodic points, then the dimensions of both sets are equal and the homeomorphism is shown to be absolutely continuous with respect to the corresponding h-dimensional packing measures. This property in turn implies the conjugating homeomorphism to be Lipschitz continuous. Additionally the existence of the scaling function is shown and a version of the rigidity theorem, expressed in terms of scaling functions, is proven. We also study the real-analytic Cantor sets for which the stronger rigidity can be shown, namely that the absolute continuity of the conjugating homeomorphism alone implies its real analyticity.},
author = {Urbański, Mariusz},
journal = {Fundamenta Mathematicae},
keywords = {conformal measure; rigidity; Lipschitz classification; parabolic Cantor sets; Cantor repellors; invariant Cantor sets; fixed parabolic points},
language = {eng},
number = {3},
pages = {241-277},
title = {Parabolic Cantor sets},
url = {http://eudml.org/doc/212194},
volume = {151},
year = {1996},
}

TY - JOUR
AU - Urbański, Mariusz
TI - Parabolic Cantor sets
JO - Fundamenta Mathematicae
PY - 1996
VL - 151
IS - 3
SP - 241
EP - 277
AB - The notion of a parabolic Cantor set is introduced allowing in the definition of hyperbolic Cantor sets some fixed points to have derivatives of modulus one. Such difference in the assumptions is reflected in geometric properties of these Cantor sets. It turns out that if the Hausdorff dimension of this set is denoted by h, then its h-dimensional Hausdorff measure vanishes but the h-dimensional packing measure is positive and finite. This latter measure can also be dynamically characterized as the only h-conformal measure. It is relatively easy to see that any two parabolic Cantor sets formed with the help of the same alphabet are canonically topologically conjugate and we then discuss the rigidity problem of what are the possibly weakest sufficient conditions for this topological conjugacy to be "smoother". It turns out that if the conjugating homeomorphism preserves the moduli of the derivatives at periodic points, then the dimensions of both sets are equal and the homeomorphism is shown to be absolutely continuous with respect to the corresponding h-dimensional packing measures. This property in turn implies the conjugating homeomorphism to be Lipschitz continuous. Additionally the existence of the scaling function is shown and a version of the rigidity theorem, expressed in terms of scaling functions, is proven. We also study the real-analytic Cantor sets for which the stronger rigidity can be shown, namely that the absolute continuity of the conjugating homeomorphism alone implies its real analyticity.
LA - eng
KW - conformal measure; rigidity; Lipschitz classification; parabolic Cantor sets; Cantor repellors; invariant Cantor sets; fixed parabolic points
UR - http://eudml.org/doc/212194
ER -

References

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