On trajectories entirely situated near a~hyperbolic set

D. V. Anosov

Contemporary Mathematics. Fundamental Directions, Proceedings of the Sixth International Conference on Differential and Functional-Differential Equations (Moscow, August 14–21, 2011). Part 1, Tome 45 (2012), pp. 5-17

Résumé

Let $I_1$ be a set of points such that their trajectories under a diffeomorphism $f_1$ are entirely close enough to a hyperbolic set $F_1$ of this diffeomorphism. Then it is proved that the structure of $I_1$ and restriction $f_1|_{I_1}$ ("motion in $I_1$") are essentially defined (up to an equivariant homeomorphism) by “internal dynamics” in $F_1$, i.e., by the restriction $f_1|_{F_1}$. (In more detail: the equivariant homeomorphism $g_1$ of the set $F_1$ on the hyperbolic set $F_2$ of the second diffeomorphism $f_2$ (probably, acting on another manifold $M_2$) is extendable to an equivariant homeomorphic embedding $I_1\to M_2$. The image of the imbedding contains all the trajectories $f_2$ close enough to $F_2$.)