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Regular version of the site

Article

Cantor Type Basic Sets of Surface A-endomorphisms

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

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.