K.J. Falconer, Wavelet transforms and order-two densities of fractals, J.Statist.Phys., 67(1992), 781-793.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....h( m) R J log jDT jd m d THE SCENERY FLOW FOR HYPERBOLIC JULIA SETS 19 and invoking the variational principle for ows again, we conclude that h S t d. The proof is complete. The next theorem now follows as a corollary. Part (iii) is due, with a di erent proof, to Falconer [11]. Theorem 4.2. For a rational map T with hyperbolic Julia set (or, more generally, for a conformal mixing repellor) i) for a.e. z 2 J , the average density at 0 2 CI of H d restricted to L z exists, and this value is a.s. constant on J . ii) If for some z 2 J , the average ....

K. Falconer, Wavelet transforms and order-two densities of fractals, J. Statistical Physics 67 (1992), 781-794.

....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 two Brownian ....

K.J. Falconer. Wavelet transforms and order{two densities of fractals. Journ. Stat. Phys., 67 (1992) 781-793.

....at x exists if D ( x) D ( x) and in this case the joint value is denoted by D ( x) Bedford and Fisher show that the average density exists almost everywhere for Hausdor measure on hyperbolic Cantor sets and zero sets of Brownian motion. Recently other authors (see for example [5], 19] and [7] have extended this result to various other classes of fractal measures with self similarity properties. Average densities have also been used for the investigation of general measures with positive lower and nite upper densities. For example, Falconer and Springer in [6] and ....

K.J. Falconer, Wavelet transforms and order-two densities of fractals, J.Statist.Phys., 67(1992), 781-793.

....D ff Gamma ( x) Average densities are also known as order two densities. Bedford and Fisher show that the lower and upper average density agree almost everywhere for Hausdorff measure on hyperbolic Cantor sets and zero sets of Brownian motion. Recently, many other authors (see for example [Fal92], FX93] and [PZ93] 1 have shown that for particular classes of fractal measures the lower and upper average densities agree. This also applies to the one sided average densities. But average densities can also be used for the investigation of more general measures. For example, Falconer and ....

K.J. Falconer. Wavelet transforms and order-two densities of fractals. Journal Stat.Physics, 67:781--793, 1992. 16

....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, particularly in the context of ....

K.J. Falconer. Wavelet transforms and order{two densities of fractals. Journ. Stat. Phys., Vol. 67, 1992, pp. 781-793.

....formula appropriated to numerical simulation. In order to enlight the main ideas of the proof of Theorem 1 we will split of it into several steps. As an auxiliary tool the Appendix contains an extension of Birkhoff s ergodic theorem which is of independent interest. It goes back to Falconer [3] in a special case. Basis works concerning notions and results summarized in section 1 and 2 are Hutchinson [6] Bedford [1] Mauldin and Urbanski [8] and Patzschke [9] A survey on the exhaustive paper [8] with detailed references may be found in Mauldin [7] 1 Conformal iterated function ....

....A motivation for this approach and measure geometric relationships between average densities and local dimensions of measures may be found in the survey paper [12] There we also refer to the former literature where special densities of this type have been treated. Closely related is the paper [3] of Falconer who proved existence and constancy of the average densities at almost all points of certain conformal repellers. The inverse mapping in the above model may be considered as the corresponding generator of the dynamical system. In order to formulate our main result we need a ....

[Article contains additional citation context not shown here]

Falconer, K. J., Wavelet transforms and order--two densities of fractals, J. Statist. Phys. 67 (1992), 781--793.

....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 case of the paths of Brownian motion in the plane, see [Mor98b] 2 ....

K.J. Falconer. Wavelet transforms and order--two densities of fractals. Journ. Stat. Phys., 67:781--793, 1992.

....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 two Brownian ....

K.J. Falconer. Wavelet transforms and ordertwo densities of fractals. Journ. Stat. Phys., 67 (1992) 781793.

....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, particularly in the context of ....

K.J. Falconer. Wavelet transforms and order{two densities of fractals. Journ. Stat. Phys., Vol. 67, 1992, pp. 781-793.

....numerical approximation. 3 Self conformal measures The model which will be chosen in this section is a generalization of the self similar case. It was first studied in Bedford [1] and is further extended in Mauldin and Urba nski [9] For the quite similar notion of conformal repellers Falconer [4] showed that the average densities of the conformal measure exist and are constant at almost all points. In the proof of our main theorem we use several notions and results from these papers. The mappings S 1 ; SN are now replaced by contracting conformal diffeomorphisms U S i (U) for ....

Falconer, K. J., Wavelet transforms and order--two densities of fractals, J. Statist. Phys. 67 (1992), 781--793.

....Densities of Self Similar Measures on the Line 103 r 0. Since it is known [Falconer 1985] that the limit does not usually exist, we seek various substitutes. Bedford and Fisher [1992] consider an average in r, which they call a second order density, and this approach has been widely investigated [Falconer 1992; Patzschke and Zahle 1993] From our point of view, this average is too crude. In the linear case, a more recent approach [Bandt 1992; Graf 1993] suggests that a much richer structure exists. In Section 4 we propose such a structure, which we call a density diagram, defined to be essentially the ....

K. J. Falconer, "Wavelet transforms and order-two densities of fractals", J. Stat. Phys. 67 (1992), 781--793.

....formula appropriated to numerical simulation. In order to enlight the main ideas of the proof of Theorem 1 we will split of it into several steps. As an auxiliary tool the Appendix contains an extension of Birkhoff s ergodic theorem which is of independent interest. It goes back to Falconer [3] in a special case. Basis works concerning notions and results summarized in section 1 and 2 are Hutchinson [6] Bedford [1] Mauldin and Urbanski [8] and Patzschke [9] A survey on the exhaustive paper [8] with detailed references may be found in Mauldin [7] 1 Conformal iterated function ....

....A motivation for this approach and measure geometric relationships between average densities and local dimensions of measures may be found in the survey paper [11] There we also refer to the former literature where special densities of this type have been treated. Closely related is the paper [3] of Falconer who proved existence and constancy of the average densities at almost all points of certain conformal repellers. The inverse mapping in the above model may be considered as the corresponding generator of the dynamical system. In order to formulate our main result we need a ....

[Article contains additional citation context not shown here]

Falconer, K. J., Wavelet transforms and order--two densities of fractals, J. Statist. Phys. 67 (1992), 781--793.

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K.J. Falconer, Wavelet transforms and order-two densities of fractals, J.Statist.Phys., 67(1992), 781-793.

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K.J. Falconer. Wavelet transforms and order--two densities of fractals. Journ. Stat. Phys., 67:781--793, 1992.

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K.J. Falconer. Wavelet transforms and order{two densities of fractals. Journ. Stat. Phys., 67 (1992) 781-793.

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K.J. Falconer. Wavelet transforms and order-two densities of fractals. Journal Stat.Physics, 67:781--793, 1992. 16

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K.J. Falconer. Wavelet transforms and order--two densities of fractals. Journ. Stat. Phys., 67 (1992) 781--793.

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K.J. Falconer. Wavelet transforms and order--two densities of fractals. Journ. Stat. Phys., 67 (1992) 781--793.

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