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Higher order local dimensions and Baire category

Lars Olsen (2011)

Studia Mathematica

Let X be a complete metric space and write (X) for the family of all Borel probability measures on X. The local dimension d i m l o c ( μ ; x ) of a measure μ ∈ (X) at a point x ∈ X is defined by d i m l o c ( μ ; x ) = l i m r ↘ 0 ( l o g μ ( B ( x , r ) ) ) / ( l o g r ) whenever the limit exists, and plays a fundamental role in multifractal analysis. It is known that if a measure μ ∈ (X) satisfies a few general conditions, then the local dimension of μ exists and is equal to a constant for μ-a.a. x ∈ X. In view of this, it is natural to expect that for a fixed x ∈ X, the local dimension...

Highly transitive subgroups of the symmetric group on the natural numbers

U. B. Darji, J. D. Mitchell (2008)

Colloquium Mathematicae

Highly transitive subgroups of the symmetric group on the natural numbers are studied using combinatorics and the Baire category method. In particular, elementary combinatorial arguments are used to prove that given any nonidentity permutation α on ℕ there is another permutation β on ℕ such that the subgroup generated by α and β is highly transitive. The Baire category method is used to prove that for certain types of permutation α there are many such possibilities for β. As a simple corollary,...

Hilbert C*-modules from group actions: beyond the finite orbits case

Michael Frank, Vladimir Manuilov, Evgenij Troitsky (2010)

Studia Mathematica

Continuous actions of topological groups on compact Hausdorff spaces X are investigated which induce almost periodic functions in the corresponding commutative C*-algebra. The unique invariant mean on the group resulting from averaging allows one to derive a C*-valued inner product and a Hilbert C*-module which serve as an environment to describe characteristics of the group action. For Lyapunov stable actions the derived invariant mean M ( ϕ x ) is continuous on X for any ϕ ∈ C(X), and the induced C*-valued...

Homeomorphism Groups and the Topologist's Sine Curve

Jan J. Dijkstra, Rachid Tahri (2010)

Bulletin of the Polish Academy of Sciences. Mathematics

It is shown that deleting a point from the topologist's sine curve results in a locally compact connected space whose autohomeomorphism group is not a topological group when equipped with the compact-open topology.

Homeomorphisms of composants of Knaster continua

Sonja Štimac (2002)

Fundamenta Mathematicae

The Knaster continuum K p is defined as the inverse limit of the pth degree tent map. On every composant of the Knaster continuum we introduce an order and we consider some special points of the composant. These are used to describe the structure of the composants. We then prove that, for any integer p ≥ 2, all composants of K p having no endpoints are homeomorphic. This generalizes Bandt’s result which concerns the case p = 2.

Homeomorphisms of inverse limit spaces of one-dimensional maps

Marcy Barge, Beverly Diamond (1995)

Fundamenta Mathematicae

We present a new technique for showing that inverse limit spaces of certain one-dimensional Markov maps are not homeomorphic. In particular, the inverse limit spaces for the three maps from the tent family having periodic kneading sequence of length five are not homeomorphic.

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