A new class of (H^k,1)-rectifiable subsets of metric spaces

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March 2013, 12(2): 881-898. doi: 10.3934/cpaa.2013.12.881

A new class of $(H^k,1)$-rectifiable subsets of metric spaces

Roberta Ghezzi ^1, and Frédéric Jean ^2,

1.	CMAP, École Polytechnique -- Team GECO, INRIA Saclay, France
2.	ENSTA ParisTech, UMA, and Team GECO, INRIA Saclay -- Île-de-France, Paris, France

Received September 2011 Revised June 2012 Published September 2012

Abstract
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The main motivation of this paper arises from the study of Carnot--Carathéodory spaces, where the class of $1$-rectifiable sets does not contain smooth non-horizontal curves; therefore a new definition of rectifiable sets including non-horizontal curves is needed. This is why we introduce in any metric space a new class of curves, called continuously metric differentiable of degree $k$, which are Hölder but not Lipschitz continuous when $k>1$. Replacing Lipschitz curves by this kind of curves we define $(H^k,1)$-rectifiable sets and show a density result generalizing the corresponding one in Euclidean geometry. This theorem is a consequence of computations of Hausdorff measures along curves, for which we give an integral formula. In particular, we show that both spherical and usual Hausdorff measures along curves coincide with a class of dimensioned lengths and are related to an interpolation complexity, for which estimates have already been obtained in Carnot--Carathéodory spaces.

Keywords: Hölder maps, Rectifiability, non Euclidean metric spaces, Hausdorff measures, sub-Riemannian geometry..

Mathematics Subject Classification: Primary: 28A78, 30L99 ; Secondary: 53C1.

Citation: Roberta Ghezzi, Frédéric Jean. A new class of $(H^k,1)$-rectifiable subsets of metric spaces. Communications on Pure and Applied Analysis, 2013, 12 (2) : 881-898. doi: 10.3934/cpaa.2013.12.881

References:

[1]	A. Agrachev, D. Barilari and U. Boscain, On the Hausdorff volume in sub-Riemannian geometry, Calc. Var. Partial Differential Equations, 43 (2012), 355-388.
[2]	L. Ambrosio and B. Kirchheim, Rectifiable sets in metric and Banach spaces, Math. Ann., 318 (2000), 527-555.
[3]	L. Ambrosio and P. Tilli, "Topics on Analysis in Metric Spaces," volume 25 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2004.
[4]	P. Assouad, Plongements lipschitziens dans $R^n$, Bull. Soc. Math. France, 111 (1983), 429-448.
[5]	A. Bellaïche, The tangent space in sub-Riemannian geometry, In "Sub-Riemannian Geometry," volume 144 of Progr. Math., pages 1-78. Birkhäuser, Basel, 1996.
[6]	A. S. Besicovitch, On the fundamental geometrical properties of linearly measurable plane sets of points, Math. Ann., 98 (1928), 422-464.
[7]	U. Boscain, G. Charlot, R. Ghezzi and M. Sigalotti, Lipschitz classification of two-dimensional almost-Riemannian distances on compact oriented surfaces, preprint, arXiv:1003.4842, to appear on Journal of Geometric Analysis, published online 23 september 2011. doi: 10.1007/s12220-011-9262-4.
[8]	E. Falbel and F. Jean, Measures of transverse paths in sub-Riemannian geometry, J. Anal. Math., 91 (2003), 231-246.
[9]	H. Federer, The $(\varphi,k)$ rectifiable subsets of $n$-space, Trans. Amer. Soc., 62 (1947), 114-192.
[10]	H. Federer, "Geometric Measure Theory," Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York, 1969.
[11]	B. Franchi, R. Serapioni and F. Serra Cassano, Rectifiability and perimeter in the Heisenberg group, Math. Ann., 321 (2001), 479-531.
[12]	J.-P. Gauthier, B. Jakubczyk and V. Zakalyukin, Motion planning and fastly oscillating controls, SIAM J. Control Optim., 48 (2009/10), 3433-3448.
[13]	J.-P. Gauthier and V. Zakalyukin, On the one-step-bracket-generating motion planning problem, J. Dyn. Control Syst., 11 (2005), 215-235.
[14]	M. Gromov, Carnot-Carathéodory spaces seen from within, In "Sub-Riemannian Geometry," volume 144 of Progr. Math., pages 79-323. Birkhäuser, Basel, 1996.
[15]	G. H. Hardy, Weierstrass's non-differentiable function, Trans. Amer. Math. Soc., 17 (1916), 301-325.
[16]	F. Jean, "Paths in Sub-Riemannian Geometry," Springer (A. Isidori, F. Lamnabhi-Lagarrigue and W. Respondek Eds.), 2000.
[17]	F. Jean, Entropy and complexity of a path in sub-Riemannian geometry, ESAIM Control Optim. Calc. Var., 9 (2003), 485-508 (electronic).
[18]	B. Kirchheim, Rectifiable metric spaces: local structure and regularity of the Hausdorff measure, Proc. Amer. Math. Soc., 121 (1994), 113-123.
[19]	V. Magnani, Characteristic points, rectifiability and perimeter measure on stratified groups, J. Eur. Math. Soc., 8 (2006), 585-609.
[20]	J. M. Marstrand, The $(\varphi, s)$ regular subsets of $n$-space, Trans. Amer. Math. Soc., 113 (1964), 369-392.
[21]	P. Mattila, "Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability," Cambridge Studies in Advanced Mathematics. Cambridge University Press, 1996.
[22]	P. Mattila, R. Serapioni and F. Serra Cassano, Characterizations of intrinsic rectifiability in Heisenberg groups, Ann. Sc. Norm. Super. Pisa Cl. Sci., 9 (2010), 687-723.
[23]	E. F. Moore, Density ratios and $(\phi,1)$ rectifiability in $n$-space, Trans. Amer. Math. Soc., 69 (1950), 324-334.
[24]	D. Preiss, Geometry of measures in $R^n$: distribution, rectifiability, and densities, Ann. of Math., 125 (1987), 537-643.
[25]	K. Weierstrass, On continuous functions of a real argument that do not have a well-defined differential quotient, G.A. Edgar, Classics on Fractals. Addison-Wesley Publishing Company, 1993.

show all references

References:

[1]	A. Agrachev, D. Barilari and U. Boscain, On the Hausdorff volume in sub-Riemannian geometry, Calc. Var. Partial Differential Equations, 43 (2012), 355-388.
[2]	L. Ambrosio and B. Kirchheim, Rectifiable sets in metric and Banach spaces, Math. Ann., 318 (2000), 527-555.
[3]	L. Ambrosio and P. Tilli, "Topics on Analysis in Metric Spaces," volume 25 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2004.
[4]	P. Assouad, Plongements lipschitziens dans $R^n$, Bull. Soc. Math. France, 111 (1983), 429-448.
[5]	A. Bellaïche, The tangent space in sub-Riemannian geometry, In "Sub-Riemannian Geometry," volume 144 of Progr. Math., pages 1-78. Birkhäuser, Basel, 1996.
[6]	A. S. Besicovitch, On the fundamental geometrical properties of linearly measurable plane sets of points, Math. Ann., 98 (1928), 422-464.
[7]	U. Boscain, G. Charlot, R. Ghezzi and M. Sigalotti, Lipschitz classification of two-dimensional almost-Riemannian distances on compact oriented surfaces, preprint, arXiv:1003.4842, to appear on Journal of Geometric Analysis, published online 23 september 2011. doi: 10.1007/s12220-011-9262-4.
[8]	E. Falbel and F. Jean, Measures of transverse paths in sub-Riemannian geometry, J. Anal. Math., 91 (2003), 231-246.
[9]	H. Federer, The $(\varphi,k)$ rectifiable subsets of $n$-space, Trans. Amer. Soc., 62 (1947), 114-192.
[10]	H. Federer, "Geometric Measure Theory," Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York, 1969.
[11]	B. Franchi, R. Serapioni and F. Serra Cassano, Rectifiability and perimeter in the Heisenberg group, Math. Ann., 321 (2001), 479-531.
[12]	J.-P. Gauthier, B. Jakubczyk and V. Zakalyukin, Motion planning and fastly oscillating controls, SIAM J. Control Optim., 48 (2009/10), 3433-3448.
[13]	J.-P. Gauthier and V. Zakalyukin, On the one-step-bracket-generating motion planning problem, J. Dyn. Control Syst., 11 (2005), 215-235.
[14]	M. Gromov, Carnot-Carathéodory spaces seen from within, In "Sub-Riemannian Geometry," volume 144 of Progr. Math., pages 79-323. Birkhäuser, Basel, 1996.
[15]	G. H. Hardy, Weierstrass's non-differentiable function, Trans. Amer. Math. Soc., 17 (1916), 301-325.
[16]	F. Jean, "Paths in Sub-Riemannian Geometry," Springer (A. Isidori, F. Lamnabhi-Lagarrigue and W. Respondek Eds.), 2000.
[17]	F. Jean, Entropy and complexity of a path in sub-Riemannian geometry, ESAIM Control Optim. Calc. Var., 9 (2003), 485-508 (electronic).
[18]	B. Kirchheim, Rectifiable metric spaces: local structure and regularity of the Hausdorff measure, Proc. Amer. Math. Soc., 121 (1994), 113-123.
[19]	V. Magnani, Characteristic points, rectifiability and perimeter measure on stratified groups, J. Eur. Math. Soc., 8 (2006), 585-609.
[20]	J. M. Marstrand, The $(\varphi, s)$ regular subsets of $n$-space, Trans. Amer. Math. Soc., 113 (1964), 369-392.
[21]	P. Mattila, "Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability," Cambridge Studies in Advanced Mathematics. Cambridge University Press, 1996.
[22]	P. Mattila, R. Serapioni and F. Serra Cassano, Characterizations of intrinsic rectifiability in Heisenberg groups, Ann. Sc. Norm. Super. Pisa Cl. Sci., 9 (2010), 687-723.
[23]	E. F. Moore, Density ratios and $(\phi,1)$ rectifiability in $n$-space, Trans. Amer. Math. Soc., 69 (1950), 324-334.
[24]	D. Preiss, Geometry of measures in $R^n$: distribution, rectifiability, and densities, Ann. of Math., 125 (1987), 537-643.
[25]	K. Weierstrass, On continuous functions of a real argument that do not have a well-defined differential quotient, G.A. Edgar, Classics on Fractals. Addison-Wesley Publishing Company, 1993.

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