Invariant sets with zero measure and full Hausdorff dimension.
Dimension product structure of hyperbolic sets.
Dimension and product structure of hyperbolic measures.
On the pointwise dimension of hyperbolic measures: a proof of the Eckmann-Ruelle conjecture.
Time weighted entropies
For invertible transformations we introduce various notions of topological entropy. For compact invariant sets these notions are all the same and equal the usual topological entropy. We show that for non-invariant sets these notions are different. They can be used to detect the direction in time in which the system evolves to highest complexity.
Diophantine classes, dimension and Denjoy maps
Dyadic Diophantine approximation and Katok's horseshoe approximation
Strongly mixing sequences of measure preserving transformations
We call a sequence of measure preserving transformations strongly mixing if tends to for arbitrary measurable , . We investigate whether one can pass to a suitable subsequence such that almost surely for all (or “many”) integrable .
Dimension of countable intersections of some sets arising in expansions in non-integer bases
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.
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