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Lowpass filters and representations of the Baumslag Solitar group
, 2004
"... Abstract. We analyze representations of the Baumslag Solitar group BS(1, N) = 〈u, t  utu −1 = t N 〉 that admit wavelets and show how such representations can be constructed from a given lowpass filter. We describe the direct integral decomposition for some examples and derive from it a general cr ..."
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Cited by 12 (4 self)
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Abstract. We analyze representations of the Baumslag Solitar group BS(1, N) = 〈u, t  utu −1 = t N 〉 that admit wavelets and show how such representations can be constructed from a given lowpass filter. We describe the direct integral decomposition for some examples and derive from it a general criterion for the existence of solutions for scaling equations. As another application, we construct a Fourier transform for some Hausdorff measures. Contents
DIOPHANTINE APPROXIMATION AND THE GEOMETRY OF LIMIT SETS IN GROMOV HYPERBOLIC METRIC SPACES
"... Abstract. In this paper, we provide a complete theory of Diophantine approximation in the limit set of a group acting on a Gromov hyperbolic metric space. This summarizes and completes what has until now been an ad hoc collection of results by many authors. In addition to providing much greater gene ..."
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Cited by 8 (5 self)
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Abstract. In this paper, we provide a complete theory of Diophantine approximation in the limit set of a group acting on a Gromov hyperbolic metric space. This summarizes and completes what has until now been an ad hoc collection of results by many authors. In addition to providing much greater generality than any prior work, our results also give new insight into the nature of the connection between Diophantine approximation and the geometry of the limit set within which it takes place. Two results are also contained here which are purely geometric: a generalization of a theorem of C. J. Bishop and P. W. Jones to Gromov hyperbolic metric spaces, and a proof that the uniformly radial limit set of a group acting on a proper geodesic Gromov hyperbolic metric space has zero PattersonSullivan measure unless the group is quasiconvexcocompact. The latter is an application of a Diophantine theorem. Contents
Nonergodic actions, cocycles and superrigidity
"... Abstract. This paper proves various results concerning nonergodic actions of locally compact groups and particularly Borel cocycles defined over such actions. The general philosophy is to reduce the study of the cocycle to the study of its restriction to each ergodic component of the action, while b ..."
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Cited by 7 (1 self)
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Abstract. This paper proves various results concerning nonergodic actions of locally compact groups and particularly Borel cocycles defined over such actions. The general philosophy is to reduce the study of the cocycle to the study of its restriction to each ergodic component of the action, while being careful to show that all objects arising in the analysis depend measurably on the ergodic component. This allows us to prove a version of the superrigidity theorems for cocycles defined over nonergodic actions. Contents
GENERALIZED CALDERÓN CONDITIONS AND REGULAR ORBIT SPACES
, 2009
"... The construction of generalized continuous wavelet transforms on locally compact abelian groups A from quasiregular representations of a semidirect product group G = A ⋊ H acting on L 2 (A) requires the existence of a squareintegrable function whose Plancherel transform satisfies Calderóntype re ..."
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Cited by 7 (1 self)
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The construction of generalized continuous wavelet transforms on locally compact abelian groups A from quasiregular representations of a semidirect product group G = A ⋊ H acting on L 2 (A) requires the existence of a squareintegrable function whose Plancherel transform satisfies Calderóntype resolution of the identity. The question then arises under what conditions such squareintegrable functions exist. The existing literature on this subject leaves a gap between sufficient and necessary criteria. In this paper, we give a characterization in terms of the natural action of the dilation group H on the character group of A. We first prove that a Calderóntype resolution of the identity gives rise to a decomposition of Plancherel measure of A into measures on the dual orbits, and then show that the latter property is equivalent to regularity conditions on the orbit space of the dual action. Thus we obtain, for the first time, sharp necessary and sufficient criteria for the existence of a wavelet inversion formula. As a byproduct and special case of our results we obtain that discrete series subrepresentations of the quasiregular representation correspond precisely to dual orbits with positive Plancherel measure and associated compact stabilizers. Only sufficiency of the conditions was previously known.
EXCHANGEABLE MEASURES FOR SUBSHIFTS
"... Abstract. Let Ω be a Borel subset of S N where S is countable. A measure is called exchangeable on Ω, if it is supported on Ω and is invariant under every Borel automorphism of Ω which permutes at most finitely many coordinates. DeFinetti’s theorem characterizes these measures when Ω = S N. We appl ..."
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Cited by 6 (2 self)
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Abstract. Let Ω be a Borel subset of S N where S is countable. A measure is called exchangeable on Ω, if it is supported on Ω and is invariant under every Borel automorphism of Ω which permutes at most finitely many coordinates. DeFinetti’s theorem characterizes these measures when Ω = S N. We apply the ergodic theory of equivalence relations to study the case Ω ̸ = S N, and obtain versions of this theorem when Ω is a countable state Markov shift, and when Ω is the collection of beta expansions of real numbers in [0, 1] (a nonMarkovian constraint). Exchangeability. DeFinetti’s theorem says that if a stochastic process {Xn}n≥1 is exchangeable, i.e. all finite permutations {X π(n)} of {Xn}n≥1 are distributed like {Xn}, then it is distributed as a mixture of i.i.d. distributions. Here is a seemingly stronger, but equivalent formulation: Let K be the collection
Relatively finite measurepreserving extensions and lifting multipliers by Rokhlin cocycles
, 2009
"... Dedicated to Stephen Smale in recognition of his contributions to topology and dynamical systems ..."
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Cited by 4 (1 self)
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Dedicated to Stephen Smale in recognition of his contributions to topology and dynamical systems
Ergodic decomposition for measures quasiinvariant under Borel actions of inductively compact groups
, 2012
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