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

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.
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. Google Scholar

[2]

L. Ambrosio and B. Kirchheim, Rectifiable sets in metric and Banach spaces,, Math. Ann., 318 (2000), 527. 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, (2004). Google Scholar

[4]

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

[5]

A. Bellaïche, The tangent space in sub-Riemannian geometry,, In, (1996), 1. Google Scholar

[6]

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

[9]

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

[10]

H. Federer, "Geometric Measure Theory,", Die Grundlehren der mathematischen Wissenschaften, (1969). Google Scholar

[11]

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

[14]

M. Gromov, Carnot-Carathéodory spaces seen from within,, In, (1996), 79. Google Scholar

[15]

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

[16]

F. Jean, "Paths in Sub-Riemannian Geometry,", Springer (A. Isidori, (2000). Google Scholar

[17]

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

[18]

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

[19]

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

[20]

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

[23]

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

[24]

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

[25]

K. Weierstrass, On continuous functions of a real argument that do not have a well-defined differential quotient,, G.A. Edgar, (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. Google Scholar

[2]

L. Ambrosio and B. Kirchheim, Rectifiable sets in metric and Banach spaces,, Math. Ann., 318 (2000), 527. 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, (2004). Google Scholar

[4]

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

[5]

A. Bellaïche, The tangent space in sub-Riemannian geometry,, In, (1996), 1. Google Scholar

[6]

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

[9]

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

[10]

H. Federer, "Geometric Measure Theory,", Die Grundlehren der mathematischen Wissenschaften, (1969). Google Scholar

[11]

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

[14]

M. Gromov, Carnot-Carathéodory spaces seen from within,, In, (1996), 79. Google Scholar

[15]

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

[16]

F. Jean, "Paths in Sub-Riemannian Geometry,", Springer (A. Isidori, (2000). Google Scholar

[17]

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

[18]

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

[19]

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

[20]

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

[23]

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

[24]

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

[25]

K. Weierstrass, On continuous functions of a real argument that do not have a well-defined differential quotient,, G.A. Edgar, (1993). Google Scholar

[1]

Daniel Genin, Serge Tabachnikov. On configuration spaces of plane polygons, sub-Riemannian geometry and periodic orbits of outer billiards. Journal of Modern Dynamics, 2007, 1 (2) : 155-173. doi: 10.3934/jmd.2007.1.155

[2]

Erlend Grong, Alexander Vasil’ev. Sub-Riemannian and sub-Lorentzian geometry on $SU(1,1)$ and on its universal cover. Journal of Geometric Mechanics, 2011, 3 (2) : 225-260. doi: 10.3934/jgm.2011.3.225

[3]

Stefan Sommer, Anne Marie Svane. Modelling anisotropic covariance using stochastic development and sub-Riemannian frame bundle geometry. Journal of Geometric Mechanics, 2017, 9 (3) : 391-410. doi: 10.3934/jgm.2017015

[4]

Yunlong Huang, P. S. Krishnaprasad. Sub-Riemannian geometry and finite time thermodynamics Part 1: The stochastic oscillator. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-26. doi: 10.3934/dcdss.2020072

[5]

Nicolas Dirr, Federica Dragoni, Max von Renesse. Evolution by mean curvature flow in sub-Riemannian geometries: A stochastic approach. Communications on Pure & Applied Analysis, 2010, 9 (2) : 307-326. doi: 10.3934/cpaa.2010.9.307

[6]

Gabriel Fuhrmann, Jing Wang. Rectifiability of a class of invariant measures with one non-vanishing Lyapunov exponent. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5747-5761. doi: 10.3934/dcds.2017249

[7]

Paul W. Y. Lee, Chengbo Li, Igor Zelenko. Ricci curvature type lower bounds for sub-Riemannian structures on Sasakian manifolds. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 303-321. doi: 10.3934/dcds.2016.36.303

[8]

Karla Díaz-Ordaz. Decay of correlations for non-Hölder observables for one-dimensional expanding Lorenz-like maps. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 159-176. doi: 10.3934/dcds.2006.15.159

[9]

Rafael De La Llave, R. Obaya. Regularity of the composition operator in spaces of Hölder functions. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 157-184. doi: 10.3934/dcds.1999.5.157

[10]

Vincent Lynch. Decay of correlations for non-Hölder observables. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 19-46. doi: 10.3934/dcds.2006.16.19

[11]

Pedro Duarte, Silvius Klein, Manuel Santos. A random cocycle with non Hölder Lyapunov exponent. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4841-4861. doi: 10.3934/dcds.2019197

[12]

Samia Challal, Abdeslem Lyaghfouri. Hölder continuity of solutions to the $A$-Laplace equation involving measures. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1577-1583. doi: 10.3934/cpaa.2009.8.1577

[13]

Eugen Mihailescu. Unstable manifolds and Hölder structures associated with noninvertible maps. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 419-446. doi: 10.3934/dcds.2006.14.419

[14]

Moisey Guysinsky, Serge Yaskolko. Coincidence of various dimensions associated with metrics and measures on metric spaces. Discrete & Continuous Dynamical Systems - A, 1997, 3 (4) : 591-603. doi: 10.3934/dcds.1997.3.591

[15]

Wei Ouyang, Li Li. Hölder strong metric subregularity and its applications to convergence analysis of inexact Newton methods. Journal of Industrial & Management Optimization, 2017, 13 (5) : 0-0. doi: 10.3934/jimo.2019105

[16]

Qingsong Gu, Jiaxin Hu, Sze-Man Ngai. Geometry of self-similar measures on intervals with overlaps and applications to sub-Gaussian heat kernel estimates. Communications on Pure & Applied Analysis, 2020, 19 (2) : 641-676. doi: 10.3934/cpaa.2020030

[17]

Meiyu Su. True laminations for complex Hènon maps. Conference Publications, 2003, 2003 (Special) : 834-841. doi: 10.3934/proc.2003.2003.834

[18]

Tien-Cuong Dinh, Nessim Sibony. Rigidity of Julia sets for Hénon type maps. Journal of Modern Dynamics, 2014, 8 (3&4) : 499-548. doi: 10.3934/jmd.2014.8.499

[19]

Charles Pugh, Michael Shub, Amie Wilkinson. Hölder foliations, revisited. Journal of Modern Dynamics, 2012, 6 (1) : 79-120. doi: 10.3934/jmd.2012.6.79

[20]

Jinpeng An. Hölder stability of diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 315-329. doi: 10.3934/dcds.2009.24.315

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]