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332
Dimensions and measures in infinite iterated function systems
 PROC. LONDON MATH. SOC
, 1996
"... The Hausdorff and packing measures and dimensions of the limit sets of iterated function systems generated by countable families of conformal contractions are investigated. Conformal measures for such systems, reflecting geometric properties of the limit set, are introduced, proven to exist, and to ..."
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Cited by 82 (23 self)
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The Hausdorff and packing measures and dimensions of the limit sets of iterated function systems generated by countable families of conformal contractions are investigated. Conformal measures for such systems, reflecting geometric properties of the limit set, are introduced, proven to exist, and to be unique. The existence of a unique invariant probability equivalent to the conformal measure is derived. Our methods employ the concepts of the PerronFrobenius operator, symbolic dynamics on an infinite dimensional shift space, and the properties of the above mentioned ergodic invariant measure. A formula for the Hausdorff dimension of the limit set in terms of the pressure function is derived. Fractal phenomena not exhibited by finite systems are shown to appear in the infinite case. In particular a variety of conditions are provided for Hausdorff and packing measures to be positive or finite, and a number of examples are described showing the appearance of various possible combinations for these quantities. One example given special attention is the limit set associated to the complex continued fraction expansion  in particular lower and upper estimates for its Hausdor dimension are given. A large natural class of systems whose limit sets are "dimensionless in the restricted sense" is described.
Approximation By Translates Of Refinable Functions
, 1996
"... . The functions f 1 (x); : : : ; fr (x) are refinable if they are combinations of the rescaled and translated functions f i (2x \Gamma k). This is very common in scientific computing on a regular mesh. The space V 0 of approximating functions with meshwidth h = 1 is a subspace of V 1 with meshwidth ..."
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Cited by 68 (14 self)
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. The functions f 1 (x); : : : ; fr (x) are refinable if they are combinations of the rescaled and translated functions f i (2x \Gamma k). This is very common in scientific computing on a regular mesh. The space V 0 of approximating functions with meshwidth h = 1 is a subspace of V 1 with meshwidth h = 1=2. These refinable spaces have refinable basis functions. The accuracy of the computations depends on p, the order of approximation, which is determined by the degree of polynomials 1; x; : : : ; x p\Gamma1 that lie in V 0 . Most refinable functions (such as scaling functions in the theory of wavelets) have no simple formulas. The functions f i (x) are known only through the coefficients c k in the refinement equationscalars in the traditional case, r \Theta r matrices for multiwavelets. The scalar "sum rules" that determine p are well known. We find the conditions on the matrices c k that yield approximation of order p from V 0 . These are equivalent to the StrangFix condition...
Distribution of zeros of random and quantum chaotic sections of positive line bundles
 Commun. Math. Phys
, 1999
"... Abstract. We study the limit distribution of zeros of certain sequences of holomorphic sections of high powers LN of a positive holomorphic Hermitian line bundle L over a compact complex manifold M. Our first result concerns ‘random ’ sequences of sections. Using the natural probability measure on t ..."
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Cited by 50 (23 self)
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Abstract. We study the limit distribution of zeros of certain sequences of holomorphic sections of high powers LN of a positive holomorphic Hermitian line bundle L over a compact complex manifold M. Our first result concerns ‘random ’ sequences of sections. Using the natural probability measure on the space of sequences of orthonormal bases {SN j} of H0 (M, LN), we show that for almost every sequence {SN j}, the associated sequence of zero currents 1 N Z S N j tends to the curvature form ω of L. Thus, the zeros of a sequence of sections sN ∈ H 0 (M, L N) chosen independently and at random become uniformly distributed. Our second result concerns the zeros of quantum ergodic eigenfunctions, where the relevant orthonormal bases {S N j} of H0 (M, L N) consist of eigensections of a quantum ergodic map. We show that also in this case the zeros become uniformly distributed. 1.
Expansive Subdynamics
, 1997
"... . This paper provides a framework for studying the dynamics of commuting homeomorphisms. Let ff be a continuous action of Z d on an infinite compact metric space. For each subspace V of R d we introduce a notion of expansiveness for ff along V , and show that there are nonexpansive subspaces in ever ..."
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Cited by 39 (8 self)
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. This paper provides a framework for studying the dynamics of commuting homeomorphisms. Let ff be a continuous action of Z d on an infinite compact metric space. For each subspace V of R d we introduce a notion of expansiveness for ff along V , and show that there are nonexpansive subspaces in every dimension d \Gamma 1. For each k d the set E k (ff) of expansive k dimensional subspaces is open in the Grassman manifold of all kdimensional subspaces of R d . Various dynamicalproperties of ff are constant, or vary nicely, within a connected component of E k (ff), but change abruptly when passing from one expansive component to another. We give several examples of this sort of "phase transition," including the topological and measuretheoretic directional entropies studied by Milnor, zeta functions, and dimension groups. For d = 2 we show that, except for one unresolved case, every open set of directions whose complement is nonempty can arise as an E 1 (ff). The unr...
Dynamical Systems on Translation Bounded Measures: PURE POINT DYNAMICAL AND DIFFRACTION SPECTRA
, 2003
"... Certain topological dynamical systems are considered that arise from actions of σcompact locally compact Abelian groups on compact spaces of translation bounded measures. Such a measure dynamical system is shown to have pure point dynamical spectrum if and only if its diffraction spectrum is pure ..."
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Cited by 38 (23 self)
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Certain topological dynamical systems are considered that arise from actions of σcompact locally compact Abelian groups on compact spaces of translation bounded measures. Such a measure dynamical system is shown to have pure point dynamical spectrum if and only if its diffraction spectrum is pure point.
Spectrum of dynamical systems arising from Delone sets
 American Math. Soc.: Providence RI
, 1997
"... We investigate spectral properties of the translation action on the orbit closure of a Delone set. In particular, sufficient conditions for pure discrete spectrum are given, based on the notion of almost periodicity. Connections with diffraction spectrum are discussed. 1 Introduction A set ae R d ..."
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Cited by 37 (2 self)
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We investigate spectral properties of the translation action on the orbit closure of a Delone set. In particular, sufficient conditions for pure discrete spectrum are given, based on the notion of almost periodicity. Connections with diffraction spectrum are discussed. 1 Introduction A set ae R d is a Delone set if there exist positive constants R and r such that every ball of radius R intersects and every ball of radius r contains at most one point of . The collection of all such sets with fixed R and r can be equipped with a metric to form a compact space. The group R d acts on this space by translations. We study the spectral properties of this action restricted to some invariant subsets. We begin with a description of eigenvalues (with continuous eigenfunctions) assuming that the restricted action is minimal. Then we consider dynamics with respect to an ergodic invariant measure and obtain sufficient conditions for the system to have pure discrete spectrum. These conditions a...
Statistical Properties of Probabilistic ContextFree Grammars
 Computational Linguistics
, 1999
"... This article proves a number of useful properties of probabilistic contextfree grammars (PCFGs). In this section, we give an introduction to the results and related topics ..."
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Cited by 36 (0 self)
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This article proves a number of useful properties of probabilistic contextfree grammars (PCFGs). In this section, we give an introduction to the results and related topics
Multiwavelets: Theory and Applications
, 1996
"... A function OE(t) is refinable if it satisfies a dilation equation OE(t) = P k C k OE(2t \Gamma k). A refinable function (scaling function) generates a multiresolution analysis (MRA): Set of nested subspaces : : : V \Gamma1 ae V 0 ae V 1 : : : such that S 1 j=\Gamma1 V j = L 2 (R), T 1 j=\Gam ..."
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Cited by 35 (4 self)
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A function OE(t) is refinable if it satisfies a dilation equation OE(t) = P k C k OE(2t \Gamma k). A refinable function (scaling function) generates a multiresolution analysis (MRA): Set of nested subspaces : : : V \Gamma1 ae V 0 ae V 1 : : : such that S 1 j=\Gamma1 V j = L 2 (R), T 1 j=\Gamma1 V j = f0g, and translates OE(t \Gamma k) constitute a basis of V 0 . Then a basis fw jk : w jk = w(2 j t \Gamma k) j; k 2 Zg of L 2 (R) is generated by a wavelet w(t), whose translates w(t \Gamma k) form a basis of W 0 , V 1 = V 0 \Phi W 0 . A standard (scalar) MRA assumes that there is only one scaling function. We make a step forward and allow several scaling functions OE 0 (t); : : : ; OE r\Gamma1 (t) to generate a basis of V 0 . The vector OE(t) = [OE 0 (t) : : : OE r\Gamma1 (t)] T satisfies a dilation equation with matrix coefficients C k . Associated with OE(t) is a multiwavelet w(t) = [w 0 (t) : : : w r\Gamma1 (t)] T . Unlike the scalar case, construction of a multiwave...