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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

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 & 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.  Google Scholar

[2]

L. Ambrosio and B. Kirchheim, Rectifiable sets in metric and Banach spaces, Math. Ann., 318 (2000), 527-555.  Google Scholar

[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.  Google Scholar

[4]

P. Assouad, Plongements lipschitziens dans $R^n$, Bull. Soc. Math. France, 111 (1983), 429-448.  Google Scholar

[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.  Google Scholar

[6]

A. S. Besicovitch, On the fundamental geometrical properties of linearly measurable plane sets of points, Math. Ann., 98 (1928), 422-464.  Google Scholar

[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.  Google Scholar

[8]

E. Falbel and F. Jean, Measures of transverse paths in sub-Riemannian geometry, J. Anal. Math., 91 (2003), 231-246.  Google Scholar

[9]

H. Federer, The $(\varphi,k)$ rectifiable subsets of $n$-space, Trans. Amer. Soc., 62 (1947), 114-192.  Google Scholar

[10]

H. Federer, "Geometric Measure Theory," Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York, 1969.  Google Scholar

[11]

B. Franchi, R. Serapioni and F. Serra Cassano, Rectifiability and perimeter in the Heisenberg group, Math. Ann., 321 (2001), 479-531.  Google Scholar

[12]

J.-P. Gauthier, B. Jakubczyk and V. Zakalyukin, Motion planning and fastly oscillating controls,, SIAM J. Control Optim., 48 (): 3433.   Google Scholar

[13]

J.-P. Gauthier and V. Zakalyukin, On the one-step-bracket-generating motion planning problem, J. Dyn. Control Syst., 11 (2005), 215-235.  Google Scholar

[14]

M. Gromov, Carnot-Carathéodory spaces seen from within, In "Sub-Riemannian Geometry," volume 144 of Progr. Math., pages 79-323. Birkhäuser, Basel, 1996.  Google Scholar

[15]

G. H. Hardy, Weierstrass's non-differentiable function, Trans. Amer. Math. Soc., 17 (1916), 301-325.  Google Scholar

[16]

F. Jean, "Paths in Sub-Riemannian Geometry," Springer (A. Isidori, F. Lamnabhi-Lagarrigue and W. Respondek Eds.), 2000.  Google Scholar

[17]

F. Jean, Entropy and complexity of a path in sub-Riemannian geometry, ESAIM Control Optim. Calc. Var., 9 (2003), 485-508 (electronic).  Google Scholar

[18]

B. Kirchheim, Rectifiable metric spaces: local structure and regularity of the Hausdorff measure, Proc. Amer. Math. Soc., 121 (1994), 113-123.  Google Scholar

[19]

V. Magnani, Characteristic points, rectifiability and perimeter measure on stratified groups, J. Eur. Math. Soc., 8 (2006), 585-609.  Google Scholar

[20]

J. M. Marstrand, The $(\varphi, s)$ regular subsets of $n$-space, Trans. Amer. Math. Soc., 113 (1964), 369-392.  Google Scholar

[21]

P. Mattila, "Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability," Cambridge Studies in Advanced Mathematics. Cambridge University Press, 1996.  Google Scholar

[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.  Google Scholar

[23]

E. F. Moore, Density ratios and $(\phi,1)$ rectifiability in $n$-space, Trans. Amer. Math. Soc., 69 (1950), 324-334.  Google Scholar

[24]

D. Preiss, Geometry of measures in $R^n$: distribution, rectifiability, and densities, Ann. of Math., 125 (1987), 537-643.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

L. Ambrosio and B. Kirchheim, Rectifiable sets in metric and Banach spaces, Math. Ann., 318 (2000), 527-555.  Google Scholar

[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.  Google Scholar

[4]

P. Assouad, Plongements lipschitziens dans $R^n$, Bull. Soc. Math. France, 111 (1983), 429-448.  Google Scholar

[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.  Google Scholar

[6]

A. S. Besicovitch, On the fundamental geometrical properties of linearly measurable plane sets of points, Math. Ann., 98 (1928), 422-464.  Google Scholar

[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.  Google Scholar

[8]

E. Falbel and F. Jean, Measures of transverse paths in sub-Riemannian geometry, J. Anal. Math., 91 (2003), 231-246.  Google Scholar

[9]

H. Federer, The $(\varphi,k)$ rectifiable subsets of $n$-space, Trans. Amer. Soc., 62 (1947), 114-192.  Google Scholar

[10]

H. Federer, "Geometric Measure Theory," Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York, 1969.  Google Scholar

[11]

B. Franchi, R. Serapioni and F. Serra Cassano, Rectifiability and perimeter in the Heisenberg group, Math. Ann., 321 (2001), 479-531.  Google Scholar

[12]

J.-P. Gauthier, B. Jakubczyk and V. Zakalyukin, Motion planning and fastly oscillating controls,, SIAM J. Control Optim., 48 (): 3433.   Google Scholar

[13]

J.-P. Gauthier and V. Zakalyukin, On the one-step-bracket-generating motion planning problem, J. Dyn. Control Syst., 11 (2005), 215-235.  Google Scholar

[14]

M. Gromov, Carnot-Carathéodory spaces seen from within, In "Sub-Riemannian Geometry," volume 144 of Progr. Math., pages 79-323. Birkhäuser, Basel, 1996.  Google Scholar

[15]

G. H. Hardy, Weierstrass's non-differentiable function, Trans. Amer. Math. Soc., 17 (1916), 301-325.  Google Scholar

[16]

F. Jean, "Paths in Sub-Riemannian Geometry," Springer (A. Isidori, F. Lamnabhi-Lagarrigue and W. Respondek Eds.), 2000.  Google Scholar

[17]

F. Jean, Entropy and complexity of a path in sub-Riemannian geometry, ESAIM Control Optim. Calc. Var., 9 (2003), 485-508 (electronic).  Google Scholar

[18]

B. Kirchheim, Rectifiable metric spaces: local structure and regularity of the Hausdorff measure, Proc. Amer. Math. Soc., 121 (1994), 113-123.  Google Scholar

[19]

V. Magnani, Characteristic points, rectifiability and perimeter measure on stratified groups, J. Eur. Math. Soc., 8 (2006), 585-609.  Google Scholar

[20]

J. M. Marstrand, The $(\varphi, s)$ regular subsets of $n$-space, Trans. Amer. Math. Soc., 113 (1964), 369-392.  Google Scholar

[21]

P. Mattila, "Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability," Cambridge Studies in Advanced Mathematics. Cambridge University Press, 1996.  Google Scholar

[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.  Google Scholar

[23]

E. F. Moore, Density ratios and $(\phi,1)$ rectifiability in $n$-space, Trans. Amer. Math. Soc., 69 (1950), 324-334.  Google Scholar

[24]

D. Preiss, Geometry of measures in $R^n$: distribution, rectifiability, and densities, Ann. of Math., 125 (1987), 537-643.  Google Scholar

[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.  Google Scholar

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