S. Graf, On Bandt's tangential distribution for self-similar measures, Monatsh.Math., 120(1995), 223-246.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of scale invariance. The idea is to take the set A not as a complete gure, but rather as a basic space from which we can sample circular views of part of A [24] One can consider centered views around points of A which lead to the so called tangential distribution of self similar measures [15] but here we study views inside V which are not centered. Details will be given in a forthcoming paper. 4 An algorithm deciding on separation Now we are back to deterministic self similar sets. The open set condition (OSC) was introduced already 1946 by Moran to prove that the Hausdor measure ....

S. Graf, On Bandt's tangential distribution for self-similar measures, Monatshefte Math. 120 (1995), 223-246

.... 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 to the ....

....by Ciesielski and Taylor (1962) ffl A further characteristic, which appears to be worth studying, is the tangent measure distribution of the occupation measures of a planar Brownian motion. For the definition and results about tangent measure distributions see for example Bandt (1990) Graf (1995) or Morters and Preiss (1997) 3 The order three density of planar Brownian motion In this section we prove Theorems 2.1 and 2.1. Let us first see, why we can restrict our attention to the behaviour of the occupation measure at the origin. An elegant approach to this problem is the idea of Palm ....

S. Graf. On Bandt's tangential distribution for self-similar measures. Mh. Math. 120 (1995) 223--246.

....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 Euclidean space and ....

....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 studied ....

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S. Graf. On Bandt's tangential distribution for self similar measures. Mh. Math., 120:223-246, 1995.

.... ( 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) and M(IR) is ....

....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 every x 2 IR de ne ....

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S. Graf, On Bandt's tangential distribution for self-similar measures, Monatsh.Math., 120(1995), 223-246.

.... a lot of papers concerning fractal properties like multifractal spectrum, fractal derivatives, Fourier transformation, and tangential distributions of self similar random measures (see for example Falconer [F] Olsen [O] Patzschke and Z ahle [PZ1] Z1] Lau and Wang [LW] Strichartz [S] Graf [G], Arbeiter and Patzschke [AP] In all these papers the investigations were restricted to random measures with compact support. A generalization to self similar random measures with noncompact support was given in [Z3] and [A1] A First consideration of fractal random measures with ....

Graf. S., On Bandt's tangential distribution for self--similar measures, preprint 1993

....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 case and get ....

....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 case and get analog results. Moreover, we compute the tangential distribution of Phi for almost all ....

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Graf, S.: On Bandt's Tangential Distribution for Self--Similar Measures, preprint 1993

....dr r log(1=r) For a large class of fractal measures possessing some self similarity the average densities of order two were shown to exist and be equal to a constant at almost every x. Examples include the natural measures on random and deterministic self similar sets, see e.g. 19] 20] [8], mixing repellers, see [5] the zero set and path of Brownian motion, see [1] 7] and intersections of Brownian paths in 3 space, see [18] It was also shown that average densities can distinguish between di erent m part Cantor sets of equal dimension, see [13] or [6] In many cases, ....

S. Graf. On Bandt's tangential distribution for self-similar measures. Mh. Math. Vol. 120, 1995, pp. 223-246.

....properties) In the sequel we will additionally assume that (C4) intX F 6= Together with (C3) this is said to be the strong open set condition. Then the logarithmic distance function to the boundary of X is integrable: Z fi fi ln dist (x; X) fi fi d(x) 1 : 10) This was proved by Graf [4] for self similar measures and in a more general context by Patzschke [9] Lemma 4.2 for q = 0) 3 The average density of the conformal measure One of the local measure geometric quantities of fractal models are the average densities of associated measures. They have been introduced in Bedford ....

....X i=1 r s i j ln r i j j Gamma1 X i6=j Z i (X) Z j (X) jy Gamma zj Gammas d(y) d(z) at almost all x. This expression (up to the constant) may be interpreted as a copy based truncated s energy integral of . Existence and constancy of this density was proved in [10] and Graf [4] presented a formula which may be transformed to the above one by means of Fubini and the invariance property of ( 5) and (6) In the general case Graf s method does not work, but our formula is similar to the self similar case when interpreting dist oe (y; z) as a time average distance ....

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Graf, S.; On Bandt's tangential distribution for self--similar measures, Monatsh.--Math. 120 (1995), 223--246.

....ne probability distributions P (x; P on M(R d ) by P (M) 1 log(1= Z 1 1 M x;r r dr r for Borel sets M M(R d ) Note that the choice of the random scales according to dr=r is well adapted to the geometry of self similar sets. The following theorem is due to Graf (1995). Theorem 1.2 Let be the canonical measure on a deterministic self similar set as above and its dimension. Then, for almost every x, the distributions P converge weakly to the distribution P of an self similar random measure, which is the canonical randomization of . 2 Tangent ....

....estimate in the present situation. The full proof of Lemma 2.2 can be found in M orters Preiss (1998) 3 Further results Some measure theoretic applications of Theorem 2. 1 are given in M orters Preiss (1998) and in M orters (1997) Examples of tangent measure distributions were studied in Graf (1995), Arbeiter Patzschke (1996) Krieg M orters (1998) and, for the case of planar Brownian paths, in M orters (1998) ....

S. Graf. On Bandt's tangential distribution for self-similar measures. Mh. Math. 120:223-246, 1995.

....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, the rescaling ....

S. Graf, On Bandt's tangential distribution for self-similar measures. Mh. Math., 120:223--246, 1995. 9

....dr r log(1=r) For a large class of fractal measures possessing some self similarity the average densities of order two were shown to exist and be equal to a constant at almost every x. Examples include the natural measures on random and deterministic self similar sets, see e.g. 19] 20] [8], mixing repellers, see [5] the zero set and path of Brownian motion, see [1] 7] and intersections of Brownian paths in 3 space, see [18] It was also shown that average densities can distinguish between di erent m part Cantor sets of equal dimension, see [13] or [6] In many cases, ....

S. Graf. On Bandt's tangential distribution for self-similar measures. Mh. Math. Vol. 120, 1995, pp. 223-246.

....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 tangent measure ....

....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 Patzschke [APa] ....

S. Graf. On Bandt's tangential distribution for self similar measures. Mh. Math., 120:223--246, 1995.

....book [3] In [12] it is shown that the average densities D s (x) exist and equal some positive constant at Gamma a:a: x. An expression for this constant in terms of the dynamical system associated with the mapping S, whose inverse branches are the S 1 ; SN , is derived in Graf [6]. The potential theoretic version reads as follows: Theorem. See [14] D s (x) 1 s X i6=j Z S i (F ) Z S j (F ) 1 jy Gamma zj s d (y) d (z) at Gamma a:a: x, where = N P i=1 r s i j log r i j. This formula may also be used for numerical approximation. 3 ....

Graf, S.; On Bandt's tangential distribution for self--similar measures, Monatsh.--Math. 120 (1995), 223--246

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S. Graf. On Bandt's tangential distribution for self similar measures. Mh. Math., 120:223-246, 1995.

....to check that X oe2 Gamma p oe = 1 (2.5) for every minimal covering Gamma. In the following we always assume that (S 1 ; SN ) satisfies the OSC. Then p (S oe (A) S (A) 0 if oe 6OE and 6OE oe and p (S oe (A) p oe (2. 6) for all oe; 2 f1; Ng (see, for instance [4], Lemma 3.3) For a 2 A let i 2 f1; Ng be the first index with a 2 S i (A) Set f(a) S Gamma1 i (a) Then f : A A; a f(a) is a Borel measurable p measure preserving transformation. It is well known that f is ergodic (see, for instance, Barnsley Demko[1] p. 261 for the ....

S. Graf, On Bandt's tangential distribution for self--similar measures, Monatshefte fur Mathematik 120 (1995), 223-246

....continuous part lim n 1 ne d n;r exists and is nite and strictly positive (see [1] This fact implies that the quantization dimension D r (P ) lim n 1 log n log en;r of such a P equals d. For self similar probabilities P satisfying the strong separation condition it was shown in [2] that there is a unique D r 2 (0; 1) such that the sequence (ne Dr n;r ) n2N is bounded and bounded away from 0 and hence that P has quantization dimension D r . In the present note we show that for self similar probabilities satisfying the open set condition the quantization dimension D r = ....

.... ; 1 : n ; A = S (A) s = 1 ; s 1 : s n ; 1 : n ; and p = 1 ; p 1 : p n ; 1 : n : If (S 1 ; SN ) satis es the OSC then P (A A ) 0 if and are incomparable and moreover, P (A ) p (see [2], Lemma 3.3) 3. Statement of the main result For r 2 (0; 1) there exists a unique D r 2 (0; 1) satisfying N X i=1 (p i s r i ) Dr r Dr = 1 (see [3] Lemma 14.4) 3 3.1 Theorem Let (S 1 ; SN ) satisfy the OSC and let P be the self similar probability corresponding to (S 1 ; ....

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S. Graf, On Bandt's tangential distribution for self{similar measures, Mh. Math. 120(1995), 223-246

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S. Graf, On Bandt's tangential distribution for self-similar measures, Monatsh.Math., 120(1995), 223-246.

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S. Graf, On Bandt's tangential distribution for self-similar measures. Mh. Math., 120:223--246, 1995.

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S. Graf. On Bandt's tangential distribution for self-similar measures. Mh. Math. 120:223-246, 1995.

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S. Graf, On Bandt's tangential distribution for self-similar measures. Mh. Math., 120:223--246, 1995. 9

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S. Graf. On Bandt's tangential distribution for self-similar measures. Mh. Math. 120, 223--246 (1995).

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S. Graf. On Bandt's tangential distribution for self-similar measures. Mh. Math., 120:223-- 246, 1995.

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S. Graf. On Bandt's tangential distribution for self-similar measures. Mh. Math. 120 (1995) 223--246.

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S. Graf, On Bandt's tangential distribution for self-similar measures. Mh. Math., 120:223--246, 1995.

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S. Graf, On Bandt's tangential distribution for self-similar measures, Monatsh. Math. 120 (1995), 223 -- 246.

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