Abstract. We introduce a new concept of dimension for metric spaces, the so-called topological Hausdorff dimension. It is defined by a very natural combination of the definitions of the topological dimension and the Hausdorff dimension. The value of the topological Hausdorff dimension is always between the topological dimension and the Hausdorff dimension, in particular, this new dimension is a non-trivial lower estimate for the Hausdorff dimension.We examine the basic properties of this new notion of dimension, compare it to other well-known notions, determine its value for some classical fractals such as the Sierpinski carpet, the von Koch snowflake curve, Kakeya sets, the trail of the Brownian motion, etc.As our first application, we generalize the celebrated result of Chayes, Chayes and Durrett about the phase transition of the connectedness of the limit set of Mandelbrot's fractal percolation process. They proved that certain curves show up in the limit set when passing a critical probability, and we prove that actually 'thick' families of curves show up, where roughly speaking the word thick means that the curves can be parametrized in a natural way by a set of large Hausdorff dimension. The proof of this is basically a lower estimate of the topological Hausdorff dimension of the limit set. For the sake of completeness, we also give an upper estimate and conclude that in the non-trivial cases the topological Hausdorff dimension is almost surely strictly below the Hausdorff dimension.Finally, as our second application, we show that the topological Hausdorff dimension is precisely the right notion to describe the Hausdorff dimension of the level sets of the generic continuous function (in the sense of Baire category) defined on a compact metric space.
In this paper we answer a question of J. Bourgain which was motivated by questions A. Bellow and H. Furstenberg. We show that the sequence fn 2 g
We solve the following counting problem for measure preserving transformations. For f ∈ L 1 + (µ), is it true that sup n
In an earlier paper we introduced a new concept of dimension for metric spaces, the so called topological Hausdorff dimension. For a compact metric space K let dim H K and dim tH K denote its Hausdorff and topological Hausdorff dimension, respectively. We proved that this new dimension describes the Hausdorff dimension of the level sets of the generic continuous function on K, namely sup{dim H f −1 (y) : y ∈ R} = dim tH K − 1 for the generic f ∈ C(K), provided that K is not totally disconnected, otherwise every non-empty level set is a singleton. We also proved that if K is not totally disconnected and sufficiently homogeneous then dim H f −1 (y) = dim tH K − 1 for the generic f ∈ C(K) and the generic y ∈ f (K). The most important goal of this paper is to make these theorems more precise.As for the first result, we prove that the supremum is actually attained on the left hand side of the first equation above, and also show that there may only be a unique level set of maximal Hausdorff dimension.As for the second result, we characterize those compact metric spaces for which for the generic f ∈ C(K) and the generic y ∈ f (K) we have dim H f −1 (y) = dim tH K − 1. We also generalize a result of B. Kirchheim by showing that if K is self-similar then for the generic f ∈ C(K) for every y ∈ int f (K) we have dim H f −1 (y) = dim tH K − 1.Finally, we prove that the graph of the generic f ∈ C(K) has the same Hausdorff and topological Hausdorff dimension as K.2010 Mathematics Subject Classification. Primary: 28A78, 28A80 Secondary: 26A99.
Suppose Λ is a discrete infinite set of nonnegative real numbers. We say that Λ is of type 1 if the series s(x) = λ∈Λ f (x+λ) satisfies a zero-one law. This means that for any non-negative measurable f : R → [0, +∞) either the convergence set C(f, Λ) = {x : s(x) < +∞} = R modulo sets of Lebesgue zero, or its complement the divergence set D(f, Λ) = {x : s(x) = +∞} = R modulo sets of measure zero. If Λ is not of type 1 we say that Λ is of type 2.In this paper we show that there is a universal Λ with gaps monotone decreasingly converging to zero such that for any open subset G⊂R one can find a characteristic function f G such that G⊂D(f G , Λ) and C(f G , Λ) = R\G modulo sets of measure zero.We also consider the question whether C(f, Λ) can contain non-degenerate intervals for continuous functions when D(f, Λ) is of positive measure.The above results answer some questions raised in a paper of Z. Buczolich, J-P. Kahane, and D. Mauldin.
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