Proyecto FONDECYT Regular “Partially Hyperbolic Systems in Low Dimensions”
This proposal is in the areas of Dynamical Systems and Smooth Ergodic Theory, more precisely it deals with various topics involving systems which have “some” hyperbolicity. We are interested in partially hyperbolic systems with low di- mensional center, one or at most two, and in area preserving non-invertible maps of the two-torus homotopic to expanding automorphisms. Related to this systems we will address various problems related to the topological and metric entropy, volume growth, Lyapunov exponents, ergodic properties and classification.
The topics proposed can be divided into three subgroups:
1. The study of the continuity of the topological entropy with respect to the system and of the metric entropy with respect to the measure for C1 partially hyperbolic systems with low dimensional center; this includes partially hyperbolic diffeomorphisms with one dimensional center and partially hyperbolic flows with two dimensional center. We expect to obtain continuity of the topological entropy and upper semicontinuity of the metric entropy function in the first case, and upper semicontinuity of the topological and metric entropies in the second case. A counterexample to the lower semicontinuity of the topological entropy will be given. Possible extensions for noncompact manifolds and flows with singularities will be considered, as well as the continuity of the volume growth of the unstable foliation. There are several methods involved: establishing some type of expansiveness, using the relations between volume growth and entropy, make use of some possible classification results. The study of the regularity of Lyapunov exponents associated to invariant bundles within families of volume preserving diffeomorphisms and applications. The main method is computing the first and second order derivatives of the Lyapunov exponents with respect to the parameters, under suitable smoothness conditions.
2. Classification-related problems for partially hyperbolic diffeomorphisms with one-dimensional center. We hope to obtain new non-dynamically coherent examples in dimension three, extending the method from Hertz-Hertz-Ures to other manifolds. It is probable that the new examples do not have invariant two-tori, this would disprove an existing conjecture in the area. We will study the existence (and eventual density) of periodic points and periodic closed orbits in the nonwandering set. We expect some kind of duality: expanding or contracting of the center leaves together with lots of (hopefully dense in the nonwandering part) periodic points, or no expansion and no contraction of the center leaves together with the existence (or again hopefully density) of closed periodic center leaves. This duality could be a step towards other classification-type results for partially hyperbolic diffeomorphisms with one dimensional center restricted to the nonwandering set, and could be very useful in applications. We will use various topological and geometrical methods from hyperbolic dynamics.
3. The study of entropy, Lyapunov exponents, with a view towards ergodic properties, for some classes of non-invertible systems. We will consider local diffeomorphisms on T2 which are the compositions of an expanding automorphism with an area preserving diffeomorphism. The later can be from the Taylor-Chirikov standard family, conservative Henon family, or some similar family. We will search some specific (large enough) parameters, which will allow us to give lower bounds on the metric entropy of these systems with respect to the Lebesgue measure. When the metric entropy becomes greater that the entropy of the linear part, this will give us by the Ruelle inequality the existence of a set of points of positive Lebesgue measure which have a negative Lyapunov exponent. This result could be viewed as establishing existence of nonzero Lyapunov exponents for a “randomized” area preserving diffeomorphism. Ergodic properties of these systems will also be explored.
Nombre del proyecto: Partially Hyperbolic Systems in Low Dimensions
Fuente de financiamiento: CONICYT
Investigador: R. Saghin
Rol: Investigador Responsable
Periodo de ejecución: 2017-2021