# Proyecto Fondecyt Regular “Partially hyperbolic and low dimensional dynamics”

## Resumen

We will consider different problems in the area of hyperbolic and partially hyperbolic diffeomor- phisms. We plan to show that the stable and unstable foliations of transitive Anosov maps have a unique transverse measure, the associated current having nonzero homology, while in the case of Anosov flows the transversal measures of the stable and unstable foliations may not be unique and the associated currents are boundaries of currents supported on the center-stable and center- unstable foliations. The methods used will be the study of the Ruelle-Sullivan currents, and the Markov partitions for the uniformly hyperbolic maps. We expect that there exists a closed form nonzero on the the unstable foliation of transitive Anosov maps (and thus it is also a volume form on Wu), and we want to investigate when this situation happens for other partially hyperbolic diffeomorphisms, because it implies that the volume growth and the topological entropy in the un- stable direction are locally constant, algebraic numbers, given by the action induced by the map on the cohomology group of the manifold. We also expect that the property of the unstable foliation having nonzero homology is similar to the quasi-isometric property, and thus it should imply that the center-stable bundle is integrable at least in some situations. We will also study the absolute continuity of the center foliations, and we hope to give new examples supporting the conjecture that generically the center foliation is not absolutely continuous. The main methods used here will be the Mane argument, and the study of the derivatives of Lyapunov exponents with respect to parameters for smooth families of partially hyperbolic diffeomorphisms. We will consider different questions related to the entropy and volume growth for partially hyperbolic diffeomorphisms: connections between them in some specific cases, behavior under perturbations (including local and volume preserving), continuity when the dimension of the center bundle is one. We will also work on the entropy conjecture for partially hyperbolic diffeomorphisms with two-dimensional center bundle, and several problems related to the better understanding of partially hyperbolic diffeomorphisms in low dimensions.

We also plan to study the invariant measures of flows on surfaces with singularities, including those on surfaces of higher genus with several saddles, and simple and generalized Cherry flows. In particular we are interested in finding the physical measures, and understanding when they are supported at fixed points. For simple Cherry flows we expect that we have a dichotomy: if the divergence at the saddle is smaller than or equal to zero, then the flow has only two finite ergodic invariant measures supported at the fixed points, and the Dirac measure at the saddle is the physical measure; in the other case we believe that there exists a third ergodic invariant measure supported on the quasi-minimal set, and this is the physical measure. Some similar results should be obtained for generalized Cherry flows, i.e. flows on T2 with several saddles and sources, for which the return map to a transversal has irrational rotation number. The main tool is the study of the invariant measures of the return map to a transversal to the flow, and the integrability of the return time with respect to these invariant measures. Different techniques from the study of interval exchange transformations and perturbations (for flows with several saddles on surfaces of higher genus), and the study of distortion and renormalization for monotone circle maps with flat intervals (for Cherry flows), will be used. We will also consider possible extensions of the results to diffeomorphisms on surfaces, to higher dimensions, and other problems in low-dimensional dynamics.

**Nombre del proyecto: ** Partially hyperbolic and low dimensional dynamics

**Código: ** 1130611

**Fuente de financiamiento: ** CONICYT

**Investigador: ** R. Saghin

**Rol: ** Investigador Responsable

**País: ** Chile

**Periodo de ejecución: ** 2013-2017