On the unstable directions and Lyapunov exponents of Anosov endomorphisms
Volume 235 / 2016
Fundamenta Mathematicae 235 (2016), 37-48
MSC: Primary 37-XX; Secondary 37D20.
DOI: 10.4064/fm92-10-2015
Published online: 23 March 2016
Abstract
Unlike in the invertible setting, Anosov endomorphisms may have infinitely many unstable directions. Here we prove, under the transitivity assumption, that an Anosov endomorphism of a closed manifold $M$ is either special (that is, every $x \in M$ has only one unstable direction), or for a typical point in $M$ there are infinitely many unstable directions. Another result is the semi-rigidity of the unstable Lyapunov exponent of a $C^{1+\alpha }$ codimension one Anosov endomorphism that is $C^1$-close to a linear endomorphism of $\mathbb {T}^n$ for $(n \geq 2).$
