Partial hyperbolicity and specification (Q2790192)

This paper considers partially hyperbolic systems in the context of Bowen's ``specification property''. A diffeomorphism on a manifold \(M\) is called partially hyperbolic if there is a DF-invariant splitting of the tangent bundle \(TM=E^s\oplus E^c\oplus E^u\) such that \(E^s\) and \(E^u\) are uniformly contracting and uniformly expanding, respectively, and at least one of \(E^s\) or \(E^u\) is nontrivial. A homeomorphism \(f\) on a compact metric space \((X,d)\) satisfies the specification property if, for every \(\varepsilon>0\), there is an integer \(N(\varepsilon)\) for which the following condition holds: if \(I_1,I_2,\dots, I_k\) are piecewise disjoint intervals of integers with \(\min\{|m-n|: m\in I_i, n\in I_j\}\geq N(\varepsilon)\) for \(i\neq j\), and if \(x_1,x_2,\dots, x_k\in X\), then there is a point \(x\in X\) such that \(d(f^i(x), f^j(x_i))\leq\varepsilon\) for \(j\in I_i\) and \(1\leq i\leq k\).NEWLINENEWLINE The specification property was introduced by Rufus Bowen in the early 1970s; it can be interpreted to mean that an arbitrary number of pieces of orbits can be glued together to get an orbit that shadows the original one.NEWLINENEWLINE The authors' main result is the following: suppose that \(f:M\to M\) is a diffeomorphism admitting a partially hyperbolic splitting \(E^s\oplus E^c\oplus E^u\), and assume that there are two hyperbolic periodic points \(p\) and \(q\) (having associated stable and unstable manifolds \(W^s\) and \(W^u\)) with either \(\dim(E^u)= \dim W^u(p)< \dim W^u(q)\) or \(\dim(E^s)= \dim W^s(q)>\dim W^s(p)\). Then \(f\) does not satisfy the specification property.NEWLINENEWLINE A corollary then gives that for \(\dim M=3\), there is a \(C^1\)-dense open subset \({\mathcal P}\) in the set of robustly nonhyperbolic diffeomorphisms so that no \(f\in{\mathcal P}\) satisfies the specification property.NEWLINENEWLINE The authors' proof of the main theorem relies in an important way on the holonomy map along the strong unstable foliation.