Cantor Type Basic Sets of Surface A-endomorphisms

Russian Journal of Nonlinear Dynamics. 2021. Vol. 17. No. 3. P. 335–345.

Grines V., Zhuzhoma E. V.

The paper is devoted to an investigation of the genus of an orientable closed surface M2
which admits A-endomorphisms whose nonwandering set contains a one-dimensional strictly
invariant contracting repeller Λr with a uniquely defined unstable bundle and with an admissible
boundary of finite type. First, we prove that, if M2 is a torus or a sphere, then M2 admits
such an endomorphism. We also show that, if Ω is a basic set with a uniquely defined unstable
bundle of the endomorphism f : M2 → M2 of a closed orientable surface M2 and f is not a
diffeomorphism, then Ω cannot be a Cantor type expanding attractor. At last, we prove that,
if f : M2 → M2 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 Ωr with a uniquely defined unstable bundle and such that the lamination consisting
of stable manifolds of Ωr is regular, then M2 is a two-dimensional torus T2 or a two-dimensional
sphere S2.

Keywords: attractor, repeller, Axiom A-endomorphisms, regular lamination