# Entropy and complexity of a path in sub-riemannian geometry

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cocv287.dvi ESAIM: Control, Optimisation and Calculus of Variations August 2003, Vol. 9, 485{508 DOI: 10.1051/cocv:2003024 ENTROPY AND COMPLEXITY OF A PATH IN SUB-RIEMANNIAN GEOMETRY Fre´de´ric Jean1 Abstract. We characterize the geometry of a path in a sub-Riemannian manifold using two metric invariants, the entropy and the complexity. The entropy of a subset A of a metric space is the minimum number of balls of a given radius ε needed to cover A. It allows one to compute the Hausdorff dimension in some cases and to bound it from above in general. We define the complexity of a path in a sub- Riemannian manifold as the infimum of the lengths of all trajectories contained in an ε-neighborhood of the path, having the same extremities as the path. The concept of complexity for paths was first developed to model the algorithmic complexity of the nonholonomic motion planning problem in robotics. In this paper, our aim is to estimate the entropy, Hausdorff dimension and complexity for a path in a general sub-Riemannian manifold. We construct first a norm ‖ · ‖ε on the tangent space that depends on a parameter ε > 0. Our main result states then that the entropy of a path is equivalent to the integral of this ε-norm along the path. As a corollary we obtain upper and lower bounds for the Hausdorff dimension of a path. Our second main result is that complexity and entropy are equivalent for generic paths. We give also a computable sufficient condition on the path for this equivalence to happen. Mathematics Subject Classification. 53C17. Received July 18, 2002. Revised February 27, 2003. 1. Introduction In a sub-Riemannian geometry, submanifolds may have a Hausdor� dimension greater than their topological dimension. What values can the Hausdor� dimension have? More generally, what are the metric properties of sub-Riemannian submanifolds? In this paper we will answer these questions for one-dimensional submanifolds, or paths, using two geometric invariants, the entropy and the complexity. The ent

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