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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 43 (8 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.
Differentiating maps into L1 and the geometry of BV functions
 Ann. Math
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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 33 (8 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
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 26 (9 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
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 25 (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.
Geometric inequalities and generalized Ricci bounds in the Heisenberg group, in "Int
 Math. Res. Not. IMRN
"... gemeinschaft getragenen Sonderforschungsbereiches 611 an der Universität Bonn entstanden und als Manuskript vervielfältigt worden. Bonn, Juli 2006 Geometric inequalities and generalized Ricci bounds in the Heisenberg group Nicolas JUILLET ..."
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Cited by 22 (0 self)
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gemeinschaft getragenen Sonderforschungsbereiches 611 an der Universität Bonn entstanden und als Manuskript vervielfältigt worden. Bonn, Juli 2006 Geometric inequalities and generalized Ricci bounds in the Heisenberg group Nicolas JUILLET
The asymptotic geometry of negatively curved spaces: uniformization, geometrization and rigidity
 ICM Proceedings
, 2006
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