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Stably ergodic dynamical systems and partial hyperbolicity. (English) Zbl 0883.58025
The authors study partially hyperbolic diffeomorphisms \(f:M \to M\) of a compact, connected, boundaryless manifold \(M\). A diffeomorphism \(f\) is partially hyperbolic if \(Tf:TM\to TM\) leaves invariant a continuous splitting \(TM= E^u \oplus E^c \oplus E^s\), where \(Tf\) expands \(E^u\), \(Tf\) contracts \(E^s\) and \(\forall p\in M\) \(\sup |T_pf |_{E^s} |< \inf m(T_p |_{E^c})\) and \(\sup|T_p f|_{E^c} |< \inf m(T_p f|_{E^u})\), where \(m(T)= \inf \{|T\nu|:|\nu |= 1\}\). Subbundles \((E,F)\) are said to have the accessibility property if every pair of points in \(M\) can be joined by a piecewise \(C^1\) path. It is said that a diffeomorphism \(f\) has sufficiently Hölder invariant bundles if the Hölder exponents of the bundles \(E^u\), \(E^c\), \(E^s\) are greater than some constant, depending on \(\dim M\), and satisfying certain conditions. If the spectra of \(T^uf\), \(T^cf\) and \(T^sf\) lie in thin, well separated annuli, it is said that \(f\) has a bunched spectrum.
Some main results of the paper:
1. If a \(C^2\), volume preserving diffeomorphism \(f:M \to M\) is partially hyperbolic, dynamically coherent, has the essential accessibility property, and its invariant bundles are sufficiently Hölder, then \(f\) is ergodic.
2. If in addition to the preceding conditions \(f\) has the accessibility property, the invariant bundles of \(f\) are \(C^1\), and the spectrum of \(Tf\) is sufficiently bunched, then \(f\) is stably ergodic, i.e., \(f\) is ergodic and so is every volume preserving diffeomorphism of \(M\) that \(C^2\) approximates it.
3. The time-one map of the geodesic flow on the unit tangent bundle \(M\) of a compact Riemannian manifold of constant negative curvature is stably ergodic.

37D99 Dynamical systems with hyperbolic behavior
37A99 Ergodic theory
Full Text: DOI
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