C. Bandt, The tangent distribution for self-similar measures, Lecture at the 5th Conference on Real Analysis and Measure Theory, Capri, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... limit theorems involving order three averages have been a subject of intensive study in probability theory, see for example Foldes (1993) or Marcus and Rosen (1995) Various refinements and generalizations of the average density approach of Bedford and Fisher were suggested, see for example Bandt (1992), Graf (1995) or Morters (1997) In this paper we study the density distributions of the occupation measure of planar Brownian motion. More precisely, we show that, with probability one, the distribution of the density function at the origin with respect to a random scale distributed according ....

C. Bandt. The tangent distribution for self-similar measures. Lecture at the 5th Conference on Real Analysis and Measure Theory, Capri 1992.

....is, loosely speaking, to attach to every point of the set a family of random measures, called the dimensional tangent measure distributions at the point, which describe asymptotically the dimensional scenery seen by an observer zooming down towards this point. This tool has been used by Bandt [Ba] and Graf [G] to study the regularity of the local geometry of self similar sets, but in this paper we show that its scope goes much beyond this situation and, in fact, it may be used to describe a strong regularity property possessed by every measure: We show that, for every measure on a ....

....and study the limiting distributions of the induced family of distributions on the set of enlargements. The limiting distributions de ne random tangent measures which are called tangent measure distributions. This concept rst appeared in a weaker form in [BeFi] and then in its full strength in [Ba] and [G] Bandt and Graf used tangent measure distributions to study the particular case of self similar sets and found that they have unique tangent measure distributions at almost all points, which they described explicitly as scaling invariant Palm measures. Scaling invariant Palm measures were ....

C. Bandt. The tangent distribution for self similar measures. Lecture at the 5th Conference on Real Analysis and Measure Theory, Capri, 1992.

.... ( x) 1=2) D ( x) and D ( x) D ( x) 1=2) D ( x) We infer the result from a more general formula, which is proved by means of a detailed study of the structure of the measure and which involves the notion of tangent measure distributions introduced by Bandt ([2]) and Graf ( 9] We show that for almost every point x the formula Z Z G( u) d (u) dP ( Z Z G(T u ; u) d (u) dP ( holds for every tangent measure distribution P of at x and all Borel functions G : M(IR) IR [0; 1) Here T u is the measure de ned by T u (E) u E) ....

....with positive lower and nite upper densities. Then at almost every point x the following equations hold D ( x) D ( x) 1=2) D ( x) and D ( x) D ( x) 1=2) D ( x) In order to get a more detailed analysis of the local geometry Bandt in [2] and Graf in [9] suggested the investigation of random tangent measures based on the same averaging principle. These random measures or, equivalently, probability distributions on the space M(IR) of nonnegative Radon measures with the vague topology are called tangent measure distributions. For ....

C. Bandt, The tangent distribution for self-similar measures, Lecture at the 5th Conference on Real Analysis and Measure Theory, Capri, 1992.

....that the Hausdorff dimension of Phi equals D 1 = E P N i=1 p i ln p i E P N i=1 p i ln Lip S i : Furthermore, we compute the so called generalized dimensions of Phi, introduced for example by Strichartz [S] Pesin [P] and Riedi [R2] cf. Theorem 5. 9) For a self similar measure Bandt [B] and Graf [G] introduced the notion of a tangential distribution. In the deterministic case Graf [G] showed that the tangential distribution is unique and the same at Phi almost all points. Moreover, he gave an explicite expression of it. In this paper we generalize the definition to the random ....

Bandt, C.: The Tangent Distribution for Self--Similar Measures. Lecture at the 5 th Conference on Real Analysis and Measure Theory, Capri 1992

....the topology generated by the mappings P 7 R F dP , F continuous and bounded. Elements of P ff ( x) are called ff dimensional tangent measure distributions of at x. It is easy to see that they are supported by the set Tan ff ( x) Tangent measure distributions were introduced by Bandt in [Ban92] and Graf in [Gra95] and used as a tool for the investigation of self similar sets. Tangent measure distributions of more general measures were investigated in [Mor95] Mor98a] and [MP98] There it was shown that they have interesting invariance properties. To describe them, define, for every 0, ....

C. Bandt, The tangent distribution for self-similar measures. Lecture at the 5th Conference on Real Analysis and Measure Theory, Capri, 1992.

....6BT, U.K. December 9, 1996 Summary. Tangent measure distributions appeared as a natural tool for the description of the regularity of the local geometry of self similar sets in Euclidean spaces first in weaker versions, as in Bedford and Fisher [BeFi] and then in their full strength, as in Bandt [Ba] or Graf [G] The results are best expressed using U. Zahle s [ZU] definition of statistical self similarity: At almost every point, the local geometry of a self similar set is, from a statistical point of view, described by a unique statistically self similar random measure. The definition of ....

....of positive reals whose expectations tend to infinity and study the limiting distributions of the induced family of distributions on the set of enlargements. The limiting distributions define random tangent measures which are called tangent measure distributions. This idea was applied by Bandt [Ba] and Graf [G] to the particular case of self similar measures. They found that self similar sets have a unique tangent measure distribution at almost all points, which they described explicitly. This technique has also been applied to statistically self similar random measures by Arbeiter and ....

C. Bandt. The tangent distribution for self similar measures. Lecture at the 5th Conference on Real Analysis and Measure Theory, Capri, 1992.

....i.e. the topology generated by the mappings P 7 R F dP , F continuous and bounded. The elements of P ff ( x) are probability distributions on the set Tan ff ( x) they are the ff dimensional tangent measure distributions of at x. Tangent measure distributions were introduced by Bandt ([Ban92]) and Graf ( Gra95] originally as a tool for the investigation of self similar sets. They have also turned out to be valuable for the study of more general measures (see [Mor96] MP96] or the thesis [Mor95] which is due to the invariance properties described in the following theorem. For every ....

C. Bandt, The tangent distribution for self-similar measures. Lecture at the 5th Conference on Real Analysis and Measure Theory, Capri, 1992.

.... Measure Distributions of Hyperbolic Cantor Sets by Daniela Krieg and Peter Morters Abstract: Tangent measure distributions were introduced by Bandt [2] and Graf [8] as a means to describe the local geometry of self similar sets generated by iteration of contractive similitudes. In this paper we study the tangent measure distributions of hyperbolic Cantor sets generated by certain contractive mappings, which are not necessarily similitudes. We ....

....of limit models of Bedford and Fisher [5] Mathematics Subject Classification: 28A80, 28A75, 58F12. Key words: Fractals, Cantor sets, tangent measure distributions, limit models. 1 Tangent Measure Distributions: Definition and Properties Tangent measure distributions were introduced by Bandt in [2] and, in the present form, by Graf in [8] They provide a measure theoretic tool to describe and understand the local geometry of sets and measures. To each point in the support of a measure, or in a set equipped with some natural measure, we assign a family of probability distributions on the ....

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C. Bandt. The tangent distribution for self-similar measures. Lecture at the 5th Conference on Real Analysis and Measure Theory, Capri, 1992.

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C. Bandt. The tangent distribution for self similar measures. Lecture at the 5th Conference on Real Analysis and Measure Theory, Capri, 1992.

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C. Bandt, The tangent distribution for self-similar measures, Lecture at the 5th Conference on Real Analysis and Measure Theory, Capri, 1992.

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C. Bandt, The tangent distribution for self-similar measures. Lecture at the 5th Conference on Real Analysis and Measure Theory, Capri, 1992.

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C. Bandt, The tangent distribution for self-similar measures. Lecture at the 5th Conference on Real Analysis and Measure Theory, Capri, 1992.

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C. Bandt. The tangent distribution for self-similar measures. Lecture at the 5th Conference on Real Analysis and Measure Theory, Capri (1992).

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C. Bandt. The tangent distribution for self-similar measures. Lecture at the 5th Conference on Real Analysis and Measure Theory, Capri, 1992.

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C. Bandt. The tangent distribution for self-similar measures. Lecture at the 5th Conference on Real Analysis and Measure Theory, Capri 1992.

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C. Bandt, The tangent distribution for self-similar measures. Lecture at the 5th Conference on Real Analysis and Measure Theory, Capri, 1992.

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