6533b7dbfe1ef96bd1270261
RESEARCH PRODUCT
Periodic measures and partially hyperbolic homoclinic classes
Jinhua ZhangJinhua ZhangChristian Bonattisubject
Pure mathematicsMathematics::Dynamical SystemsGeneral MathematicsClosure (topology)Dynamical Systems (math.DS)01 natural sciencespartial hyperbolicityquasi-hyperbolic stringBlenderFOS: Mathematicsnon-hyperbolic measureErgodic theoryHomoclinic orbitMathematics - Dynamical Systems0101 mathematics[MATH]Mathematics [math]ergodic measureperiodic measureMathematicsfoliationsTransitive relationApplied MathematicsMSC (2010): Primary 37D30 37C40 37C50 37A25 37D25010102 general mathematicsRegular polygonTorusstabilityFlow (mathematics)systemsDiffeomorphismrobust cycleLyapunov exponentdescription
In this paper, we give a precise meaning to the following fact, and we prove it: $C^1$-open and densely, all the non-hyperbolic ergodic measures generated by a robust cycle are approximated by periodic measures. We apply our technique to the global setting of partially hyperbolic diffeomorphisms with one dimensional center. When both strong stable and unstable foliations are minimal, we get that the closure of the set of ergodic measures is the union of two convex sets corresponding to the two possible $s$-indices; these two convex sets intersect along the closure of the set of non-hyperbolic ergodic measures. That is the case for robustly transitive perturbation of the time one map of a transitive Anosov flow, or of the skew product of an Anosov torus diffeomorphism by rotation of the circle.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2019-04-18 |