### ?

## Cantor Type Basic Sets of Surface A-endomorphisms

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.