T. Bedford and A.M. Fisher, Analogues of the Lebesgue density theorem for fractal sets of reals and integers, Proc.London Math. Soc.(3), 64(1992), 95-124.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....in nature (clouds, soil, Swiss cheese and other porous materials . are built by choice of random mappings. 3 Axiomatic approach to self similarity The author has suggested an axiomatic approach to random self similarity [4,5] which uses ideas of Mandelbrot, U. Z ahle and several other authors [25,9,19,24]. Let B denote the unit ball in IR d : A random measure on B is a function from a certain probability space into the set M 1 (B) of probability measures on B: For our purpose, we do not need this function. We shall work with the distribution of the random variable which is a probability ....

T. Bedford and A.M. Fisher, Analogues of the Lebesgue density theorem for fractal sets of reals and integers, Proc. London Math. Soc. (3) 64 (1992), 95-124

.... the lower and upper hull of r (B(x; r) It is natural to try and describe the oscillation between the lower and upper hull and also nd a suitable average value for (B(x; r) A rst step in this direction is the investigation of the average densities introduced by Bedford and Fisher in [BF92], see also [KF97] for an introduction. For certain fractal measures Bedford and Fisher observed that, although d (r) does not converge to a nonnegative limit, it is possible to de ne a generalized limit using classical summation techniques of Hardy and Riesz. This generalized limit de nes an ....

.... #0 1 log(1= Z 1 (B(x; r) r) dr r : For many fractal measures this limit was shown to exist for gauge functions of the type (r) r . Examples include Hausdor measures on deterministic and random self similar sets, mixing repellers or occupation measures of stable processes, see [BF92], PZ94] KF92] and [FX95] We remark that average densities were also used to characterize geometric regularity of sets, see [FS95] JM96] PM97] or symmetry properties of measures, see [M98a] MP98] Our rst result shows that for the intersection local time measure on the intersection of ....

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T. Bedford and A.M. Fisher. Analogues of the Lebesgue density theorem for fractal sets of reals and integers. Proc. London Math. Soc.(3). 64 (1992) 95-124.

....two. AMS Subject Classification: 28A75, 28A80, 60G17, 60J65. Keywords: Brownian motion, occupation measure, average density, logarithmic averages, density distribution, pathwise Kallianpur Robbins law. 1 Introduction In order to study the fine local properties of fractal sets and measures Bedford and Fisher (1992) introduced a range of average densities of different orders. Whereas the classical densities fail to exist for fractal measures, see for example Mattila (1995) the average densities of order two were shown to exist for a wide range of fractal measures, like for example deterministic and random ....

.... fail to exist for fractal measures, see for example Mattila (1995) the average densities of order two were shown to exist for a wide range of fractal measures, like for example deterministic and random self similar sets, mixing repellers or random measures related to stable processes, see Bedford and Fisher (1992), Patzschke and M. Zahle (1993) Falconer (1992) and Falconer and Xiao (1995) Average densities were also used to characterize geometric regularity of sets, see Falconer and Springer (1994) Marstrand (1996) Morters (1998) or symmetry properties of measures, see Morters (1997) For the class of ....

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T. Bedford and A.M. Fisher. Analogues of the Lebesgue density theorem for fractal sets of reals and integers. Proc. London Math. Soc.(3). 64 (1992) 95--124.

....measures may be rather large and contain measures that do not have much better geometric properties than the original one. Therefore for the investigation of non recti able measures more re ned tools seem to be necessary. One class of tools is based on an averaging idea of Bedford and Fisher in [BeFi]. Instead of looking directly at all limit points of the enlargements of about x as the enlargement factor goes to in nity, we take a natural family of probability distributions on the set of positive reals whose mass tends to in nity and study the limiting distributions of the induced family of ....

....reals whose mass tends to in nity 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 ....

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T. Bedford and A.M. Fisher. Analogues of the Lebesgue density theorem for fractal sets of reals and integers. Proc. London Math. Soc.(3), 64:95-124, 1992.

....Properties of Average Densities and Tangent Measure Distributions of Measures on the Line Peter M orters Universit at Kaiserslautern, Fachbereich Mathematik, 67663 Kaiserslautern, Germany. E Mail: peter mathematik.uni kl.de Abstract: Answering a question by Bedford and Fisher in [4] we show that for the circular and one sided average densities of a Radon measure on the line with positive lower and nite upper densities the following relations hold almost everywhere D ( x) D ( x) 1=2) D ( x) and D ( x) D ( x) 1=2) D ( ....

....every x. Examples of measures ful lling these conditions are Hausdor measures on many sets including self similar sets and statistically self similar sets, measures arising in dynamical systems theory and many more. Typically these measures do not have obvious self similarity properties. In [4] Bedford and Fisher introduce average or order two densities for the study of measures of fractional dimension. For these measures the density functions t 7 ( x t; x t] t uctuate as t tends to 0 and therefore the limit does not exist (see [3] Bedford and Fisher apply a logarithmic ....

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T. Bedford and A.M. Fisher, Analogues of the Lebesgue density theorem for fractal sets of reals and integers, Proc.London Math. Soc.(3), 64(1992), 95-124.

....Local Degree of Differentiability y By M. Z ahle (Received 00.00.0000) 1. Introduction In the last ten years in fractal geometry a considerable number of strong mathematical tools has been developed. In particular, there are some attempts in a geometricanalytical direction. Bedford and Fisher [4] introduced the concept of fractional density for Cantor type sets which was explicitly computed in Patzschke and Zahle [18] for the classical middle third Cantor set. The density function may also be interpreted as a kind of fractional derivative of the associated Lebesgue function, the so called ....

T. Bedford and A. Fisher, Analogues of the Lebesgue density theorem for fractal sets of reals and integers, Proc. London Math. Soc. 64 (1992), 95-124.

....that the lower one sided average densities of these measures do not vanish, and therefore the concept of average densities is able to reveal some of the local symmetry a measure with finite and positive densities necessarily possesses. Average densities were introduced by Bedford and Fisher in [BF92] for the study of measures of fractional dimension. For these measures the density functions t 7 ( x Gamma t; x t] t ff fluctuate as t tends to 0 and therefore the limit does not exist. Bedford and Fisher apply a logarithmic average and define the lower and upper circular average densities ....

T. Bedford and A. Fisher. Analogues of the Lebesgue density theorem for fractal sets of reals and integers. Proc.London.Math. Soc.(3), 64:95--124, 1992.

....it is important to study parameters which go beyond the measurement of size and characterize ner features of the set, like its local density or its geometric regularity. Not many such parameters are established in fractal geometry, the notion of average density introduced by Bedford and Fisher in [1] is one of the most popular concepts and it has given rise to a good deal of recent publications, see for example [6] and references therein. A striking example of two important random sets with the same exact Hausdor dimension gauge are the path of a Brownian motion on the one hand and the ....

....occur. The rst problem consists in the fact that D(x) cannot be de ned as lim r 0 (B(x; r) r , as this limit fails to exist for all irregular measures and the function oscillates as r # 0 (see [22] or [15] for a precise statement of this fact) To handle this oscillation, Bedford and Fisher [1] suggested to use an averaging method based on classical summation techniques of Hardy and Riesz. For n 2 they de ne the average density of order n of at x as lim k 1 1 k Z k 0 (B(x; 1= exp (n 1) a) 1= exp (n 1) a) da ; where exp (n) is the nth iterate of the exponential ....

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T. Bedford and A.M. Fisher. Analogues of the Lebesgue density theorem for fractal sets of reals and integers. Proc. London Math. Soc.(3), Vol. 64, 1992, pp. 95-124. 23

....of F . It agrees with the normalized s dimensional Hausdorff measure restricted to F , in particular, s = dimH F . We are interested in the average s density of given by D s (x) lim 0 1 j ln j 1 Z Gamma B(x; r) Delta r s 1 r dr : It was introduced by Bedford and Fisher [2] in connection with non existence of ordinary densities for fractal measures. Relationships between these densities and (local) Hausdorff dimension are discussed in the survey paper [12] Having in mind the potential theoretic interpretation of Hausdorff dimension by capacity dimension we present ....

....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 and Fisher [2] in order to improve the infinitesimal scaling procedure, since ordinary densities do not exist for fractional dimensions. Here we are interested in the s conformal measure mentioned above and its average density of order s = dimH F at x 2 X. It is defined by D s (x) lim 0 1 j ln j ....

Bedford,T., and Fisher, A. M., Analogues of the Lebesgue density theorem for fractal sets of reals and integers, Proc. London Math. Soc. 64 (1992), 95--124.

....the proof. The results of Marstrand and Preiss imply that for measures of fractal nature the lower and upper densities cannot coincide. As it would be very desirable to describe the lacunarity of a fractal measure about its points by means of a density parameter, Bedford and Fisher introduce in [BF92] the notion of average densities into fractal geometry. Applying classical summation techniques of Hardy and Riesz to the density functions they define the lower and upper average ff densities of order two of at x as D ff ( x) lim inf #0 1 log(1= Z 1 (U(x; r) r ff dr r ; ....

....ff ( x) d ff ( x) Several authors have shown that for measures with some self similarity properties the lower and upper average densities in the appropriate dimension coincide and define an interesting fractal parameter. Examples include the natural measures on hyperbolic Cantor sets, see [BF92], random self similar sets, see [PZ94] mixing repellers, see [KF92] paths of stable processes, see [FX95] and intersections of Brownian paths in IR 3 , see [NS98] However there are also natural examples of measures where this approach fails to give a density parameter, for example in the ....

T. Bedford and A.M. Fisher, Analogues of the Lebesgue density theorem for fractal sets of reals and integers. Proc. London Math. Soc.(3), 64:95--124, 1992.

....describe its lower and upper hull. It is natural to try and describe the oscillation between the lower and upper hull and also nd a suitable average value for (B(x; r) A rst step in this direction is the investigation of the average densities introduced by Bedford and Fisher in [BF92], see also [KF97] for an introduction. For certain fractal measures Bedford and Fisher observed that, although d (r) does not converge to a nonnegative limit, it is possible to de ne a generalized limit using classical summation techniques of Hardy and Riesz. This generalized limit de nes an ....

.... #0 1 log(1= Z 1 (B(x; r) r) dr r : For many fractal measures this limit was shown to exist for gauge functions of the type (r) r ff . Examples include Hausdoroe measures on deterministic and random selfsimilar sets, mixing repellers or occupation measures of stable processes, see [BF92], PZ94] KF92] and [FX95] We remark that average densities were also used to characterize geometric regularity of sets, see [FS95] JM96] PM97] or symmetry properties of measures, see [M98a] MP98] Our rst result shows that for the intersection local time measure on the intersection of ....

[Article contains additional citation context not shown here]

T. Bedford and A.M. Fisher. Analogues of the Lebesgue density theorem for fractal sets of reals and integers. Proc. London Math. Soc.(3). 64 (1992) 95124.

....it is important to study parameters which go beyond the measurement of size and characterize ner features of the set, like its local density or its geometric regularity. Not many such parameters are established in fractal geometry, the notion of average density introduced by Bedford and Fisher in [1] is one of the most popular concepts and it has given rise to a good deal of recent publications, see for example [6] and references therein. A striking example of two important random sets with the same exact Hausdor dimension gauge are the path of a Brownian motion on the one hand and the ....

....occur. The rst problem consists in the fact that D(x) cannot be de ned as lim r 0 (B(x; r) r , as this limit fails to exist for all irregular measures and the function oscillates as r # 0 (see [22] or [15] for a precise statement of this fact) To handle this oscillation, Bedford and Fisher [1] suggested to use an averaging method based on classical summation techniques of Hardy and Riesz. They de ne the average density of order n of at x as lim k 1 1 k Z k 0 (B(x; 1= exp (n 1) a) 1= exp (n 1) a) da ; where exp (n) is the nth iterate of the exponential function. ....

[Article contains additional citation context not shown here]

T. Bedford and A.M. Fisher. Analogues of the Lebesgue density theorem for fractal sets of reals and integers. Proc. London Math. Soc.(3), Vol. 64, 1992, pp. 95-124.

....that a.e. scenery ow is (up to rotation) the same now follows from ergodicity of ( J 1 ; with respect to this measure. For the same reason (now with no need to worry about rotations) one has, by the Birkho ergodic theorem, an analogue of the Lebesgue Density Theorem for the set J (see [3], 4] 5] 12] and [13] there is a constant c 0 (the average density, also previously called the order two density) such that for a.e. z 2 J , lim T 1 1 T Z T 0 B(z; e t ) e td dt exists and equals c (see Theorem 4.2) This is an average density of the Hausdor measure, ....

....Hausdor dimension of J is not equal to 1, one always has equality of dimension and entropy, and moreover, that the continuous semiconjugacy between the model scenery ow to the full scenery ow is at most nite to one. One can also construct a scenery ow at a point x in e.g. a Brownian zero set [3], the middle third set and more generally a hyperbolic C 1 Cantor set [4, 3] and in a Fuchsian or Kleinian limit set [13] Scenery ows of certain families of circle di eomorphisms are studied in [2] To make the transition from [3] 12] to the present perspective, note that the scaling ow ....

[Article contains additional citation context not shown here]

T. Bedford and A.M. Fisher, Analogues of the Lebesgue density theorem for fractal sets of reals and integers, Proc. London Math. Soc. 64 (1992), 95-124.

....that for the fractal case (HD(J) 6= 1) one always has equality of dimension and entropy, and moreover, that the continuous semiconjugacy between the model scenery ow to the full scenery ow is at most nite to one. One can also construct a scenery ow at a point x in e.g. a Brownian zero set [BF1], the middle third set and more generally a hyperbolic C 1 Cantor set [BF2, 3] and in a Fuchsian or Kleinian limit set [F2] Scenery ows of certain families of circle di eomorphisms are studied in [AF] To make the transition from [BF1] F1] to the present perspective, note that the ....

.... a scenery ow at a point x in e.g. a Brownian zero set [BF1] the middle third set and more generally a hyperbolic C 1 Cantor set [BF2, 3] and in a Fuchsian or Kleinian limit set [F2] Scenery ows of certain families of circle di eomorphisms are studied in [AF] To make the transition from [BF1], F1] to the present perspective, note that the scaling ow on local times with local uniform topology corresponds exactly to the scenery ow on sets with the measure topology) For the present example of a hyperbolic Julia set, the dynamics of the map T : J J is used in studying the scenery ....

[Article contains additional citation context not shown here]

T. Bedford and A.M. Fisher, Analogues of the Lebesgue density theorem for fractal sets of reals and integers, Proc. London Math. Soc. 64 (1992), 95-124.

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T. Bedford and A.M. Fisher, Analogues of the Lebesgue density theorem for fractal sets of reals and integers, Proc.London Math. Soc.(3), 64(1992), 95-124.

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T. Bedford and A.M. Fisher, Analogues of the Lebesgue density theorem for fractal sets of reals and integers. Proc. London Math. Soc.(3), 64:95--124, 1992.

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T. Bedford and A.M. Fisher, Analogues of the Lebesgue density theorem for fractal sets of reals and integers. Proc. London Math. Soc.(3), 64:95--124, 1992.

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T. Bedford and A.M. Fisher. Analogues of the Lebesgue density theorem for fractal sets of reals and integers. Proc. London Math. Soc.(3). 64 (1992) 95-124.

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T. Bedford and A.M. Fisher. Analogues of the Lebesgue density theorem for fractal sets of reals and integers. Proc. London Math. Soc.(3). 64, 95--124 (1992).

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T. Bedford and A.M. Fisher. Analogues of the Lebesgue density theorem for fractal sets of reals and integers. Proc. London Math. Soc.(3), 64:95--124, 1992.

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T. Bedford and A.M. Fisher. Analogues of the Lebesgue density theorem for fractal sets of reals and integers. Proc. London Math. Soc.(3). 64 (1992) 95--124.

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T. Bedford and A. Fisher. Analogues of the Lebesgue density theorem for fractal sets of reals and integers. Proc.London.Math. Soc.(3), 64:95--124, 1992.

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T. Bedford and A.M. Fisher. Analogues of the Lebesgue density theorem for fractal sets of reals and integers. Proc. London Math. Soc.(3). 64 (1992) 95--124.

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T. Bedford and A.M. Fisher. Analogues of the Lebesgue density theorem for fractal sets of reals and integers. Proc. London Math. Soc.(3), 64:95-124, 1992.

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T. Bedford and A.M. Fisher, Analogues of the Lebesgue density theorem for fractal sets of reals and integers. Proc. London Math. Soc.(3), 64:95--124, 1992.

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