Conjugação topológica de fluxos

Abstract

In this work we present some results related to topological conjugacy and equivalence for linear differential equations type ?x = Ax, where A ? gl(d, R) and x ? R d. And posteriorly we generalize them to affine differential equations of type ?x = Ax+a, where (A, a) ? gl(d, R)×R d and x ? R d. As main results we show that if A and B are hyperbolic matrices the associated flows with ?x = Ax (or ?x = Ax + a) and ?x = Bx (or ?x = Bx + b) are topologically conjugate if and only if the stable subspaces A and B have the same dimensions. Also, we study topological conjugacy and equivalence in the torus n-dimensional. In this case, we found that the topological conjugacy (equivalence) of two nonvanishing vector fields on the torus induced by X, Y ? R n, depends on the existence of an isomorphism A: R n ? R n, Z n invariant such that A(X) = Y (A(X) = ?Y, for some ? ? Z). In the case where X, Y ? Z n, the conjugacy depends on the greatest common divisor of the entries of X and Y