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In this paper we study the relationship of the Brouwer degree of a vectorfield with the dynamics of the induced flow. Analogous relations are studiedfor the index of a vector field. We obtain new forms of the Poincar% \'{e}-Hopftheorem and of the Borsuk and Hirsch antipodal theorems. As an application, wecalculate the Brouwer degree of the vector field of the Lorenz equations inisolating blocks of the Lorenz strange set.

Shape index, Brouwer degree and Poincar\'e-Hopf theorem