Multiplicity, regularity and lipschitz geometry of real analytic hypersurfaces

This paper is devoted to studying the Lipschitz geometry of real analytic sets. We prove that the relative multiplicities are bi-Lipschitz invariant, Lipschitz regular analytic curves are C-1 and we give some counterexamples of some theorems that hold true in the complex case. Moreover, we prove that the multiplicity mod 2 of real analytic surfaces is invariant under bi-Lipschitz homeomorphisms and that the multiplicity mod 2 of real analytic hypersurfaces is invariant under image arc-analytic bi-Lipschitz homeomorphisms, which generalize some results proved by G. Valette. Finally, we prove the real version of Ephraim-Trotman's Theorem about the differentiable invariance of the multiplicity of real analytic hypersurfaces.