On one-dimensional basic sets of endomorphisms of 2-torus

In 1967, Smale [1] proposed a method for constructing nontrivial basis sets based on Anosov diffeomorphism. In fact, Smale schematically described a surgery operation on the Anosov diffeomorphism, which results in a DA-diffeomorphism with basis sets having a topological dimension one less than the dimension of an ambient manifold.
The idea of surgery is the following. In the small neighborhood of a fixed saddle point the saddle fixed point is replaced by three fixed points: two saddles and one node. In this case, the following two scenarios are equally possible: 1) the node is a source; 2) the node is a sink.
In the first case, the nonwandering set of the DA-diffeomorphism consists of a source fixed point and one-dimensional expanding attractor. In the second case the nonwandering set consists of a sink fixed point and one-dimensional contracting repeller.
In [2] a class of not one-to-one smooth maps generalizing Anosov diffeomorphisms was introduced. It seems quite natural to consider surgery operation for algebraic endomorphisms, for example, for Anosov endomorphism $f_0:\mathbb T^2\rightarrow\mathbb T^2$ given by the formula
\begin{equation} \left\{\begin{aligned}\overline x&=3x+y\\\overline y&=x+y\end{aligned}\right.\quad\text{mod }1. \end{equation}

Since $f_0$ is a local diffeomorphism, both scenarios of a surgery operation are formally possible.
In [3], the second scenario of Smale’s surgery operation for algebraic Anosov endomorphism of type (1) was numerically implemented. As a result, the $A$-endomorphism $f:\mathbb T^2\rightarrow\mathbb T^2$ of a two-dimensional torus was constructed. Its nonwandering set consists of a hyperbolic contracting repeller $\Lambda$ and one sink fixed point $O$.
The main result of this report is that the first scenario for Anosov endomorphism (1) does not lead to an $A$-endomorphism with a nontrivial one-dimensional expanding attractor having similar properties [4].
Theorem. Let $f:\mathbb T^2\rightarrow\mathbb T^2$ be $A$-endomorphism, which is $k$-fold covering, $k\geqslant2$. Then $f$ can not have one-dimensional attractor $\Lambda$, with the following properties:

• $\Lambda$ is strictly invariant;
• unstable manifold $W^u(x)$ of each point $x\in\Lambda$ does not depend on the trajectory through point $x$ and forms a one-dimensional curve, which is not closed;
• the following equality holds $\bigcup_{x\in\Lambda}W^u(x)=\Lambda$, and $\Lambda$ forms a lamination locally homeomorphic to the product of a Cantor set with an interval;
• boundary accessible from within of any connected component of the set $\mathbb T^2\setminus\Lambda$ consists of a finite number of leafs of $\Lambda$.

Acknowledgments. Research is done with financial support of Russian Science Foundation (project 17-11-01041).
Bibliography
[1] Smale S. Differentiable dynamical systems //Bulletin of the American mathematical Society. 1967, Vol. 73., No. 6., pp. 747–817.
[2] Przytycki F. Anosov endomorphisms //Studia mathematica. 1976, Vol. 3., no. 58., pp. 249–285.
[3] Kurenkov E. D. On existence of of endomorphism of 2-torus with strictly invariant contracting repeller // Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva.  2017,  Vol.19, no. 1. pp. 60–66.
[4] Grines V.Z., Zhuzhoma E.V., Kurenkov E.D. Surgery operation for Anosov endomorphisms of 2-torus does not lead to expanding attractor // Dynamical systems. 2018, Vol. 8(36), no. 3, pp. 235–244.