On existence of an endomorphism of $2$-torus with strictly invariant repeller

In the article we construct endomorphism $f$ of 2-torus. This endomorphism satisfies an axiom $A$ and has non-wondering set that contains one-dimensional contracting repeller satisfying following properties:
1) $f(\Lambda)= \Lambda$, $f^{-1}(\Lambda)= \Lambda$;
2) $\Lambda$ is locally homeomorphic to the product of the Cantor set and the interval;
3) $T^2\setminus\Lambda$ consist of a countable family of disjoint open disks.
The key idea of construction consists in applying the surgery introduced by S. Smale [1] to an algebraic Anosov endomorphism of the two-torus. We present the results of computational experiment that demonstrate correctness of our construction. Suggested construction reveals significant difference between one-dimensional basic sets of endomorphismsand one-dimensional basic sets of corresponding diffoemorphisms. In particular, the result contrasts with the fact that wondering set of axiom $A$-satisfying diffeomorphism consists of a finite number of open disks in case of spaciously situated basic set [2].