Multiplicity, regularity and blow-spherical equivalence of real analytic sets

This article is devoted to studying multiplicity and regularity of real analytic sets. We present an equivalence for real analytic sets, named blow-spherical equivalence, which generalizes differential equivalence and subanalytic bi-Lipschitz equivalence and, with this approach, we obtain several applications on analytic sets. On regularity, we show that blow-spherical regularity of real analytic implies $C^1$ smoothness only in the case of real analytic curves. On multiplicity, we present a generalization for Gau-Lipman's Theorem about differential invariance of the multiplicity in the complex and real cases, we show that the multiplicity ${\rm mod}\,2$ is invariant by blow-spherical homeomorphisms in the case of real analytic curves and surfaces and also for a class of real analytic foliations and is invariant by (image) arc-analytic blow-spherical homeomorphisms in the case of real analytic hypersurfaces, generalizing some results proved by G. Valette. We present also a complete classification of the germs of real analytic curves.