Maximizing measures for partially hyperbolic systems with compact center leaves (Q2908160)

The existence and uniqueness (finiteness) of measures of maximal entropy is an important topic in smooth ergodic theory and is well known for classical hyperbolic dynamical systems. Recently there have been many advances in this direction for systems beyond uniformly hyperbolic ones. In this paper the authors study the corresponding properties of some partially hyperbolic systems and get many interesting results.NEWLINENEWLINEMore precisely, let \(M\) be a closed 3-dimensional manifold and \(f\in\mathrm{PH}^{1+\alpha}(M)\) be an accessible and dynamical coherent partially hyperbolic diffeomorphism with compact center leaves. As noticed by the authors, the existence of measures of maximal entropy is already known in this case. See also the very recently work of \textit{L. Gang}, \textit{M. Viana} and \textit{J. Yang} [``Entropy conjecture for diffeomorphisms away from tangencies'', Preprint, \url{arXiv:1012.0514}], where it is proved that the entropy map is upper semi-continuous for all maps far from homoclinic tangency. They provided a very short proof of the existence with different flavors. Then the authors show the following interesting dichotomy:NEWLINENEWLINE(1) either there is only one such measure (say \(\mu\)): then the central Lyapunov exponent of \(\mu\) must be zero, and \((f,\mu)\) is topologically conjugate to an isometric extension of a Bernoulli shift;NEWLINENEWLINE(2) or there are more than one such measures (say \(\mu_i\), \(i=1,\dots,k\)): then every \(\mu_i\) is a hyperbolic measure and the finite extension of a Bernoulli shift. Moreover at least one \(\mu_i\) has negative central Lyapunov exponent, and at least one \(\mu_j\) has positive central Lyapunov exponent.NEWLINENEWLINEThen observing that the first alternative excludes the existence of any hyperbolic periodic points, whose complement is \(C^1\)-open and \(C^\infty\)-dense, they conclude that the second alternative happens open-and-densely among their systems. As pointed out in Remark 2, their methods can also be applied to similar systems on manifolds of any dimension.NEWLINENEWLINEThey also obtain some interesting results about 3-D nilmanifolds. For example, they show that any partially hyperbolic diffeomorphism on a 3-D nilmanifold satisfies all the conditions and hence their dichotomy.