Sequence of Induced Hausdorff Metrics on Lie Groups

Abstract

Let ϕ : G × (M, d) → (M, d) be a left action of a Lie group on a differentiable manifold endowed with a metric d, which is compatible with its topology. Let X be a compact subset of M . Then the isotropy subgroup of X is defined as H X := {g ∈ G; gX = X} and it is closed in G. The induced Hausdorff metric is a metric on the left coset manifold G/H X definedSuppose that ϕ is transitive and that there exist p ∈ M such that H X = Hp.Then gH X → gp is a diffeomorphism that identifies G/H X and M . In this work we define a discrete dynamical system of metrics on M . Let d 1 =d X , whered X stands for the intrinsic metric induced by d X . We can iterate the process on ϕ : G×(M ≡ G/H X , d 1 ) → (M ≡ G/H X , d 1 ), in order to get d 2 , d 3 and so on. We study the particular case where M = G, ϕ : G × (G, d) → (G, d) is the usual product, d is bounded above by a right invariant intrinsic metric on G and X is a finite subset of G containing the identity element. We prove that d i converges pointwise to a metric d ∞ . In addition, if d is complete and the semigroup generated by X is dense in G, then d ∞ is the distance function of a right invariant C 0 -Carnot-Carathéodory-Finsler metric. The case where d ∞ is C 0 -Finsler is studied in detail.

Geodesic fields for Pontryagin type C0-Finsler manifolds

“…In [20,21], the first author of this work and his collaborators used the term C 0 -Finsler structure for the second definition of Finsler structure. The term C 0 -Finsler structure given in Definition 2.3 coincides with the third definition of Finsler structure above.…”

Mollifier smoothing of $C^0$-Finsler structures

“…Remark 1.1. In [4] and [12], the first author and his collaborators used the term C 0 -Finsler structure for a function F : T M → R such that F (x, •) : T x M → R is a norm. In this work the term C 0 -Finsler structure is modified in order to work with non-symmetric structures.…”