The destruction of the Smale-Williams solenoids

In the paper, one represents the family of diffeomorphisms $f_{\nu}: S^3\to S^3$, $-1\leq\nu\leq 1$, depending smoothly on the parameter $\nu$ such that 1) given any $-1\leq\nu <0$, the non-wandering set of $f_{\nu}$ consists of one-dimensional expanding attractor and one-dimensional contracting repeller that are Smale-Williams solenoid; 2) the diffeomorphism $f_0$ has a non-wandering set consisting of the two zero-dimensional transitive invariant sets $\Lambda_1$ and $\Lambda_2$ such that each is homeomorphic to the product of Cantor sets, and the restriction $f_0|_{\Lambda_1\cup\Lambda_2}$ is a partially hyperbolic diffeomorphism; 3) given any $0<\nu\leq 1$, the non-wandering set of $f_{\nu}$ consists of two hyperbolic zero-dimensional transitive invariant sets each is homeomorphic to the product of Cantor sets.