Classification of systems with center-stable tori. (English) Zbl 1440.37039

Hammerlindl, Andy; Potrie, Rafael

Mich. Math. J. 68, No. 1, 147-166 (2019).

Let \(f\) be a partially hyperbolic diffeomorphism of a closed oriented 3-manifold \(M\) with a splitting \(E^s\oplus E^c\oplus E^u\) of the tangent bundle \(TM\). A center-stable (center-unstable) torus is an embebbed torus tangent to \(E^s\oplus E^c\) (respectively, to \(E^c\oplus E^u\)). It is known that there exists a finite and pairwise disjoint collection \(T_1,\dots,T_n\) of all center-stable and center-unstable tori, such that any connected component \(U\) of \(M\setminus(T_1\cup\dots\cup T_n)\) is homeomorphic to \(\mathbf{T}^2\times(0,1)\), and for any such component \(U\) there exists a \(k\geq 1\) such that \(f^k\) maps \(U\) to itself.
The authors show that for any such component \(U\) there is an embedding \(h:\;U\to\mathbf{T}^2\times\mathbf{R}\) such that the homeomorphism \(h\circ f^k\circ h^{-1}\) has the form \[ h\circ f^k\circ h^{-1}(v,x)=(A(v),\phi(v,x)), \] where \(A:\mathbf{T}^2\times\mathbf{T}^2\) is a hyperbolic toral automorphism and \(\phi:\;h(U)\to\mathbf{R}\) is continuous.

Reviewer: Sergei Yu. Pilyugin (St. Petersburg)

MSC:

37D30	Partially hyperbolic systems and dominated splittings
37D10	Invariant manifold theory for dynamical systems
37C15	Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems

Keywords:

partial hyperbolicity; embedded torus

References:

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