DCDS-S
Note on homology of expanding foliations
Radu Saghin
This note contains some remarks about the homologies that can be associated to a foliation which is invariant and uniformly expanded by a diffeomorphism. We construct a family of 'dynamical' closed currents supported on the foliation which help us relate the geometric volume growth of the leaves under the diffeomorphism with the map induced on homology in the case when these currents have nonzero homology.
keywords: invariant foliations Ruelle-Sullivan currents expanding dynamics.
DCDS
On the number of ergodic minimizing measures for Lagrangian flows
Radu Saghin
We give an example where for an open set of Lagrangians on the n-torus there is at least one cohomology class c with at least n different ergodic c-minimizing measures. One of the problems posed by Ricardo Mañé in his paper 'Generic properties and problems of minimizing measures of Lagrangian systems' (Nonlinearity, 1996) was the following:
    Is it true that for generic Lagrangians every minimizing measure is uniquely ergodic?
   A weaker statement is that for generic Lagrangians every cohomology class has exactly one minimizing measure, which of course will be ergodic. Our example shows that this can't be true and as a consequence one can hope to prove at most that for a generic Lagrangian, for every cohomology class there are at most n corresponding ergodic minimizing measures, where n is the dimension of the first cohomology group.
keywords: Action-minimizing measures. Lagrangian systems
DCDS
Volume growth and entropy for $C^1$ partially hyperbolic diffeomorphisms
Radu Saghin
We show that the metric entropy of a $C^1$ diffeomorphism with a dominated splitting and with the dominating bundle uniformly expanding is bounded from above by the integrated volume growth of the dominating (expanding) bundle plus the maximal Lyapunov exponent from the dominated bundle multiplied by its dimension. We also discuss different types of volume growth that can be associated with an expanding foliation and relationships between them, specially when there exists a closed form non-degenerate on the foliation. Some consequences for partially hyperbolic diffeomorphisms are presented.
keywords: volume growth entropy. Partially hyperbolic diffeomorphisms

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