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Dimension of a measure

Pertti Mattila, Manuel Morán, José-Manuel Rey (2000)

Studia Mathematica

We propose a framework to define dimensions of Borel measures in a metric space by formulating a set of natural properties for a measure-dimension mapping, namely monotonicity, bi-Lipschitz invariance, (σ-)stability, etc. We study the behaviour of most popular definitions of measure dimensions in regard to our list, with special attention to the standard correlation dimensions and their modified versions.

Dimension of countable intersections of some sets arising in expansions in non-integer bases

David Färm, Tomas Persson, Jörg Schmeling (2010)

Fundamenta Mathematicae

We consider expansions of real numbers in non-integer bases. These expansions are generated by β-shifts. We prove that some sets arising in metric number theory have the countable intersection property. This allows us to consider sets of reals that have common properties in a countable number of different (non-integer) bases. Some of the results are new even for integer bases.

Dimension of weakly expanding points for quadratic maps

Samuel Senti (2003)

Bulletin de la Société Mathématique de France

For the real quadratic map P a ( x ) = x 2 + a and a given ϵ > 0 a point x has good expansion properties if any interval containing x also contains a neighborhood  J of x with P a n | J univalent, with bounded distortion and B ( 0 , ϵ ) P a n ( J ) for some n . The ϵ -weakly expanding set is the set of points which do not have good expansion properties. Let α denote the negative fixed point and M the first return time of the critical orbit to [ α , - α ] . We show there is a set of parameters with positive Lebesgue measure for which the Hausdorff dimension of...

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