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Cantor Type Basic Sets of Surface $A$-endomorphisms

2021, Vol. 17, no. 3, pp.  335-345

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 $\mathbb{S}^2$.
Keywords: $A$-endomorphism, regular lamination, attractor, repeller, strictly invariant set
Citation: Grines V. Z., Zhuzhoma E. V., Cantor Type Basic Sets of Surface $A$-endomorphisms, Rus. J. Nonlin. Dyn., 2021, Vol. 17, no. 3, pp.  335-345
DOI:10.20537/nd210307