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28
Which Values of the Volume Growth and Escape Time Exponent Are Possible for a Graph?
, 2001
"... Let \Gamma = (G; E) be an infinite weighted graph which is Ahlfors ffregular, so that there exists a constant c such that c , where V (x; r) is the volume of the ball centre x and radius r. Define the escape time T (x; r) to be the mean exit time of a simple random walk on \Gamma starting at ..."
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Cited by 24 (3 self)
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Let \Gamma = (G; E) be an infinite weighted graph which is Ahlfors ffregular, so that there exists a constant c such that c , where V (x; r) is the volume of the ball centre x and radius r. Define the escape time T (x; r) to be the mean exit time of a simple random walk on \Gamma starting at x from the ball centre x and radius r. We say \Gamma has escape time exponent fi ? 0 if there exists a constant c such that c T (x; r) cr for r 1. Well known estimates for random walks on graphs imply that ff 1 and 2 fi 1 + ff.
On the differentiation of Lipschitz maps from metric measure spaces to Banach spaces
, 2006
"... Abstract. We consider metric measure spaces satisfing a doubling condition and a Poincaré inequality in the upper gradient sense. We show that the results of [Che99] on differentiability of real valued Lipschitz functions and the resulting biLipschitz nonembedding theorems for finite dimensional ve ..."
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Cited by 13 (2 self)
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Abstract. We consider metric measure spaces satisfing a doubling condition and a Poincaré inequality in the upper gradient sense. We show that the results of [Che99] on differentiability of real valued Lipschitz functions and the resulting biLipschitz nonembedding theorems for finite dimensional vector space targets extend to Banach space targets having what we term a good finite dimensional approximation. This class of targets includes separable dual spaces. We also observe that there is a straightforward extension of Pansu’s differentiation theory for Lipschitz maps between Carnot groups, [Pan89], to the most general possible class of Banach space targets, those with the RadonNikodym property. Contents
NONSMOOTH CALCULUS
"... Abstract. We survey recent advances in analysis and geometry, where first order differential analysis has been extended beyond its classical smooth settings. Such studies have applications to geometric rigidity questions, but are also of intrinsic interest. The transition from smooth spaces to singu ..."
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Cited by 12 (0 self)
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Abstract. We survey recent advances in analysis and geometry, where first order differential analysis has been extended beyond its classical smooth settings. Such studies have applications to geometric rigidity questions, but are also of intrinsic interest. The transition from smooth spaces to singular spaces where calculus is possible parallels the classical development from smooth functions to functions with weak or generalized derivatives. Moreover, there is a new way of looking at the classical geometric theory of Sobolev functions that is useful in more general contexts. 1.
SubRiemannian vs. Euclidean dimension comparison and fractal geometry on Carnot groups
, 2007
"... We solve Gromov’s dimension comparison problem for Hausdorff and box counting dimension on Carnot groups equipped with a CarnotCarathéodory metric and an adapted Euclidean metric. The proofs use sharp covering theorems relating optimal mutual coverings of Euclidean and CarnotCarathéodory balls, ..."
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Cited by 11 (8 self)
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We solve Gromov’s dimension comparison problem for Hausdorff and box counting dimension on Carnot groups equipped with a CarnotCarathéodory metric and an adapted Euclidean metric. The proofs use sharp covering theorems relating optimal mutual coverings of Euclidean and CarnotCarathéodory balls, and elements of subRiemannian fractal geometry associated to horizontal selfsimilar iterated function systems on Carnot groups. Inspired by Falconer’s work on almost sure dimensions of Euclidean selfaffine fractals we show that CarnotCarathéodory selfsimilar fractals are almost surely horizontal. As a consequence we obtain explicit dimension formulae for invariant sets of Euclidean iterated function systems of polynomial type. Jet space Carnot groups
Generalized differentiation and biLipschitz nonembedding
 in L 1 . C.R.A.S Paris, (5):297–301
, 2006
"... Résumé. We consider Lipschitz mappings f: X → V, where X is a doubling metric measure space which satisfies a Poincaré inequality, and V is a Banach space. We show that earlier differentiability and biLipschitz nonembedding results for maps f: X → R N remain valid when R N is replaced by any separa ..."
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Cited by 8 (1 self)
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Résumé. We consider Lipschitz mappings f: X → V, where X is a doubling metric measure space which satisfies a Poincaré inequality, and V is a Banach space. We show that earlier differentiability and biLipschitz nonembedding results for maps f: X → R N remain valid when R N is replaced by any separable dual space. We exhibit spaces which biLipschitz embed in L 1, but not in any separable dual V. For certain domains, including the Heisenberg group with its CarnotCaratheodory metric, we establish a new notion of differentiability for maps into L 1. This implies that the Heisenberg group does not biLipschitz embed in L 1, thereby proving a conjecture of J. Lee and A. Naor. When combined with their work, this has implications for theoretical computer science. Rubrique: Analyse. Titre: Différentiation généralisée et impossibilité d’un plongement bilipschitzien dans L 1 Résumé: Nous considérons des applications lipchitziennes f: X → V, où X est un espace métrique mesuré tel que l’on contrôle le volume des boules par doublement du rayon et qui satisfait à une inégalité de Poincaré, et où V est un espace de Banach. On montre que des résultats antérieurs de différentiabilité et de non plongement bilipschitzien pour des applications f: X → R N restent valables quand on suppose que V est un dual séparable. Nous donnons des exemples d’espaces plongés de manière bilipschitzienne dans L 1, mais qui ne sont plongeables dans aucun dual séparable. Pour certains espaces, dont le groupe d’Heisenberg muni de la métrique de CarnotCaratheodory, on établit une nouvelle notion de différentiabilité pour des applications dans L 1. Ceci implique que le groupe de Heisenberg ne possède aucun plongement bilipschitzien dans L 1, un résultat conjecturé par J. Lee et A. Naor. Quand il est combiné avec des résultats de ces deux auteurs, notre travail a des applications en informatique théorique.
Metric and geometric quasiconformality in Ahlfors regular Loewner spaces
 Conf. Geom. Dynam
"... Abstract. Recent developments in geometry have highlighted the need for abstract formulations of the classical theory of quasiconformal mappings. We modify Pansu’s generalized modulus to study quasiconformal geometry in spaces with metric and measuretheoretic properties sufficiently similar to Eucl ..."
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Cited by 7 (0 self)
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Abstract. Recent developments in geometry have highlighted the need for abstract formulations of the classical theory of quasiconformal mappings. We modify Pansu’s generalized modulus to study quasiconformal geometry in spaces with metric and measuretheoretic properties sufficiently similar to Euclidean space. Our basic objects of study are locally compact metric spaces equipped with a Borel measure which is AhlforsDavid regular of dimension Q>1, and satisfies the Loewner condition of HeinonenKoskela. For homeomorphisms between open sets in two such spaces, we prove the equivalence of three conditions: a version of metric quasiconformality, local quasisymmetry and geometric quasiconformality. We derive from these results several corollaries. First, we show that the Loewner condition is a quasisymmetric invariant in locally compact Ahlfors regular spaces. Next, we show that a proper Qregular Loewner space, Q>1, is not quasiconformally equivalent to any subdomain. (In the Euclidean case, this result is due to Loewner.) Finally, we characterize products of snowflake curves up to quasisymmetric/biLipschitz equivalence: two such products are biLipschitz equivalent if and only if they are isometric and are quasisymmetrically equivalent if and only if they are conformally equivalent.
Graphs between the elliptic and parabolic Harnack inequalities.
"... We present graphs that satisfy the uniform elliptic Harnack inequality, for harmonic functions, but not the stronger parabolic one, for solutions of the discrete heat equation. It is known from [9] that the parabolic Harnack inequality is equivalent to the conjunction of a volume regularity and a L ..."
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Cited by 7 (0 self)
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We present graphs that satisfy the uniform elliptic Harnack inequality, for harmonic functions, but not the stronger parabolic one, for solutions of the discrete heat equation. It is known from [9] that the parabolic Harnack inequality is equivalent to the conjunction of a volume regularity and a L 2 Poincaré inequality. The first example of graph satisfying the elliptic but not the parabolic Harnack inequality is due to M. Barlow and R. Bass, see [2]. It satisfies the volume regularity and not the Poincaré inequality. We construct another example that does not satisfy the volume regularity.