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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.

MSC:
 37D99 Dynamical systems with hyperbolic behavior 37A99 Ergodic theory
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