Finiteness of partially hyperbolic attractors with one-dimensional center. (English) Zbl 1453.37026
Let \(f\) be a diffeomorphism of a smooth manifold \(M\). A nonempty compact \(f\)-invariant set \(K\subset M\) is called a quasi-attractor of \(f\) if:
(i) \(K\) admits a basis of open neighborhoods \(U\) such that \(f(\overline{U})\subset U\);
(ii) \(K\) is chain-transitive.
Denote by \(PH_{c=1}^1(M)\) the set of \(C^1\)-diffeomorphisms of \(M\) that preserve a partially hyperbolic decomposition \(TM=E^s\oplus E^c\oplus E^u\) with one-dimensional \(E^c\) and nonempty \(E^s\) and \(E^u\).
The authors prove that there exists an open and dense subset \(O\) of \(PH_{c=1}^1(M)\) such that any \(f\in O\) has finitely many quasi-attractors. This result is refined for three-dimensional manifolds.
(i) \(K\) admits a basis of open neighborhoods \(U\) such that \(f(\overline{U})\subset U\);
(ii) \(K\) is chain-transitive.
Denote by \(PH_{c=1}^1(M)\) the set of \(C^1\)-diffeomorphisms of \(M\) that preserve a partially hyperbolic decomposition \(TM=E^s\oplus E^c\oplus E^u\) with one-dimensional \(E^c\) and nonempty \(E^s\) and \(E^u\).
The authors prove that there exists an open and dense subset \(O\) of \(PH_{c=1}^1(M)\) such that any \(f\in O\) has finitely many quasi-attractors. This result is refined for three-dimensional manifolds.
Reviewer: Sergei Yu. Pilyugin (St. Petersburg)
MSC:
37D05 | Dynamical systems with hyperbolic orbits and sets |
37D10 | Invariant manifold theory for dynamical systems |
37D30 | Partially hyperbolic systems and dominated splittings |
37C20 | Generic properties, structural stability of dynamical systems |
37C70 | Attractors and repellers of smooth dynamical systems and their topological structure |
37C05 | Dynamical systems involving smooth mappings and diffeomorphisms |