Vortices over Riemann surfaces and dominated splittings

We associate a flow $\phi $ with a solution of the vortex equations on a closed oriented Riemannian 2-manifold $(M,g)$ of negative Euler characteristic and investigate its properties. We show that $\phi $ always admits a dominated splitting and identify special cases in which $\phi $ is Anosov. In particular, starting from holomorphic differentials of fractional degree, we produce novel examples of Anosov flows on suitable roots of the unit tangent bundle of $(M,g)$ .

Orbital shadowing property on chain transitive sets for generic diffeomorphisms

Let f : M → M be a diffeomorphism on a closed smooth n(≥ 2) dimensional manifold M. We show that C1 generically, if a diffeomorphism f has the orbital shadowing property on locally maximal chain transitive sets which admits a dominated splitting then it is hyperbolic.

Random entropy expansiveness for diffeomorphisms with dominated splittings

We study the local entropy of typical infinite Bowen balls in random dynamical systems, and show the random entropy expansiveness for [Formula: see text] partially hyperbolic diffeomorphisms with multi one-dimensional centers. Moreover, we consider [Formula: see text] diffeomorphism [Formula: see text] with dominated splitting [Formula: see text] such that [Formula: see text] for every [Formula: see text], and all the Lyapunov exponents are non-negative along [Formula: see text] and non-positive along [Formula: see text], we prove the asymptotically random entropy expansiveness for [Formula: see text].

On the stable ergodicity of Berger–Carrasco’s example

We prove the stable ergodicity of an example of a volume-preserving, partially hyperbolic diffeomorphism introduced by Berger and Carrasco in [Berger and Carrasco. Non-uniformly hyperbolic diffeomorphisms derived from the standard map. Comm. Math. Phys.329 (2014), 239–262]. This example is robustly non-uniformly hyperbolic, with a two-dimensional center; almost every point has both positive and negative Lyapunov exponents along the center direction and does not admit a dominated splitting of the center direction. The main novelty of our proof is that we do not use accessibility.

Julia sets of complex Hénon maps

There are two natural definitions of the Julia set for complex Hénon maps: the sets [Formula: see text] and [Formula: see text]. Whether these two sets are always equal is one of the main open questions in the field. We prove equality when the map acts hyperbolically on the a priori smaller set [Formula: see text], under the additional hypothesis of substantial dissipativity. This result was claimed, without using the additional assumption, in [J. E. Fornæss, The julia set of hénon maps, Math. Ann. 334(2) (2006) 457–464], but the proof is incomplete. Our proof closely follows ideas from [J. E. Fornæss, The julia set of hénon maps, Math. Ann. 334(2) (2006) 457–464], deviating at two points, where substantial dissipativity is used. We show that [Formula: see text] also holds when hyperbolicity is replaced by one of the two weaker conditions. The first is quasi-hyperbolicity, introduced in [E. Bedford and J. Smillie, Polynomial diffeomorphisms of [Formula: see text]. VIII. Quasi-expansion. Amer. J. Math. 124(2) (2002) 221–271], a natural generalization of the one-dimensional notion of semi-hyperbolicity. The second is the existence of a dominated splitting on [Formula: see text]. Substantially dissipative, Hénon maps admitting a dominated splitting on the possibly larger set [Formula: see text] were recently studied in [M. Lyubich and H. Peters, Structure of partially hyperbolic hénon maps, ArXiv e-prints (2017)].