Cantor Type Basic Sets of Surface $A$-endomorphisms
2021, Vol. 17, no. 3, pp. 335-345
Author(s): Grines V. Z., Zhuzhoma E. V.
This work is licensed under a Creative Commons Attribution-NoDerivs 3.0 Unported License
Author(s): Grines V. Z., Zhuzhoma E. V.
The paper is devoted to an investigation of the genus of an orientable closed surface $M^2$
which admits $A$-endomorphisms whose nonwandering set contains a one-dimensional strictly
invariant contracting repeller $\Lambda_r^{}$ with a uniquely defined unstable bundle and with
an admissible boundary of finite type. First, we prove that, if $M^2$ is a torus or a
sphere, then $M^2$ admits such an endomorphism. We also show that, if $ \Omega$ is a basic set with a uniquely defined unstable bundle of the endomorphism $f\colon M^2\to M^2$ of a closed orientable surface $M^2$ and $f$ is not a diffeomorphism, then $ \Omega$ cannot be a Cantor type expanding attractor. At last, we prove that, if $f\colon M^2\to M^2$ is an $A$-endomorphism whose nonwandering set consists of a finite number of isolated periodic sink orbits and a one-dimensional strictly invariant contracting repeller of Cantor type $\Omega_r^{}$ with a uniquely defined unstable bundle and such that the lamination consisting of stable manifolds of $\Omega_r^{}$ is regular, then $M^2$ is a two-dimensional torus $\mathbb{T}^2$ or a two-dimensional sphere $\mathbb{S}^2$.
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This work is licensed under a Creative Commons Attribution-NoDerivs 3.0 Unported License