On local structure of one-dimensional basic sets of non-reversible A-endomorphisms of surfaces

V. Z. Grines ; E. V. Zhuzhoma

Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 22 (2020) no. 4, pp. 424-433.

Résumé

Recently the authors of the article discovered a meaningful class of non-reversible endomorphisms on a two-dimensional torus. A remarkable property of these endomorphisms is that their non-wandering sets contain nontrivial one-dimensional strictly invariant hyperbolic basic sets (in the terminology of S. Smale and F. Pshetitsky) which have the uniqueness of an unstable one-dimensional bundle. It was proved that nontrivial (other than periodic isolated orbits) invariant sets can only be repellers. Note that this is not the case for reversible endomorphisms (diffeomorphisms). In the present paper, it is proved that one-dimensional expanding uniquely hyperbolic and strictly invariant one-dimensional expanding attractors and one-dimensional contracting repellers of non-reversible A-endomorphisms of closed orientable surfaces have the local structure of the product of an interval by a zero-dimensional closed set (finite or Cantor). This result contrasts with the existence of one-dimensional fractal repellers arising in complex dynamics on the Riemannian sphere and not possessing the properties of the existence of a single one-dimensional unstable bundle.

Détail

Keywords: non-reversible A-endomorphisms, hyperbolic basic set, endomorphisms of surfaces.

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