We show that robustly transitive endomorphisms of a closed manifold must have a non-trivial dominated splitting or be a local diffeomorphism. This allows to get some topological obstructions for the existence of robustly transitive endomorphisms. To obtain the result, we must understand the structure of the kernel of the differential and the recurrence to the critical set of the endomorphism after perturbation.

We extend Katok’s result on ‘the approximation of hyperbolic measures by horseshoes’ to Banach cocycles. More precisely, let f be a $C^r(r>1)$ diffeomorphism of a compact Riemannian manifold M, preserving an ergodic hyperbolic measure $\mu $ with positive entropy, and let $\mathcal {A}$ be a Hölder continuous cocycle of bounded linear operators acting on a Banach space $\mathfrak {X}$. We prove that there is a sequence of horseshoes for f and dominated splittings for $\mathcal {A}$ on the horseshoes, such that not only the measure theoretic entropy of f but also the Lyapunov exponents of $\mathcal {A}$ with respect to $\mu $ can be approximated by the topological entropy of f and the Lyapunov exponents of $\mathcal {A}$ on the horseshoes, respectively. As an application, we show the continuity of sub-additive topological pressure for Banach cocycles.

For typical cocycles over subshifts of finite type, we show that for any given orbit segment, we can construct a periodic orbit such that it shadows the given orbit segment and that the product of the cocycle along its orbit is a proximal linear map. Using this result, we show that suitable assumptions on the periodic orbits have consequences over the entire subshift.

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 $(M,g)$ .

For differentiable dynamical systems with dominated splittings, we give upper estimates on the measure-theoretic tail entropy in terms of Lyapunov exponents. As our primary application, we verify the upper semi-continuity of metric entropy in various settings with domination.

A diffeomorphism $f$ has a heterodimensional cycle if there are (transitive) hyperbolic sets $\varLambda$ and $\varSigma$ having different indices (dimension of the unstable bundle) such that the unstable manifold of $\varLambda$ meets the stable one of $\varSigma$ and vice versa. This cycle has co-index $1$ if $\mathop{\mathrm{index}}(\varLambda)=\mathop{\mathrm{index}}(\varSigma)\pm1$. This cycle is robust if, for every $g$ close to $f$, the continuations of $\varLambda$ and $\varSigma$ for $g$ have a heterodimensional cycle.

We prove that any co-index $1$ heterodimensional cycle associated with a pair of hyperbolic saddles generates $C^1$-robust heterodimensioal cycles. Therefore, in dimension three, every heterodimensional cycle generates robust cycles.

We also derive some consequences from this result for $C^1$-generic dynamics (in any dimension). Two of such consequences are the following. For tame diffeomorphisms (generic diffeomorphisms with finitely many chain recurrence classes) there is the following dichotomy: either the system is hyperbolic or it has a robust heterodimensional cycle. Moreover, any chain recurrence class containing saddles having different indices has a robust cycle.

We consider a real-analytic compact surface diffeomorphism $f$, for which the tangent space over the non-wandering set $\varOmega$ admits a dominated splitting. We study the dynamical determinant

$$ d_f(z)=\exp-\sum_{n\ge1}\frac{z^n}{n}\sum_{x\in\textrm{Fix}^*f^n}|\textrm{Det}(Df^n(x)-\textrm{Id})|^{-1}, $$

where $\textrm{Fix}^*f^n$ denotes the set of fixed points of $f^n$ with no zero Lyapunov exponents. By combining previous work of Pujals and Sambarino on $C^2$ surface diffeomorphisms with, on the one hand, results of Rugh on hyperbolic analytic maps and, on the other, our two-dimensional version of the same author’s analysis of one-dimensional analytic dynamics with neutral fixed points, we prove that $d_f(z)$ is either an entire function or a holomorphic function in a (possibly multiply) slit plane.

AMS 2000 Mathematics subject classification: Primary 37C30; 37D30; 37E30