For $\unicode{x3bb}>1$, we consider the locally free ${\mathbb Z}\ltimes _\unicode{x3bb} \mathbb R$ actions on $\mathbb T^2$. We show that if the action is $C^r$ with $r\geq 2$, then it is $C^{r-\epsilon }$-conjugate to an affine action generated by a hyperbolic automorphism and a linear translation flow along the expanding eigen-direction of the automorphism. In contrast, there exists a $C^{1+\alpha }$-action which is semi-conjugate, but not topologically conjugate to an affine action.

In this paper, we study the centralizer of a partially hyperbolic diffeomorphism on ${\mathbb T}^3$ which is homotopic to an Anosov automorphism, and we show that either its centralizer is virtually trivial or such diffeomorphism is smoothly conjugate to its linear part.

We consider a smooth area-preserving Anosov diffeomorphism $f\colon \mathbb T^2\rightarrow \mathbb T^2$ homotopic to an Anosov automorphism L of $\mathbb T^2$ . It is known that the positive Lyapunov exponent of f with respect to the normalized Lebesgue measure is less than or equal to the topological entropy of L, which, in addition, is less than or equal to the Lyapunov exponent of f with respect to the probability measure of maximal entropy. Moreover, the equalities only occur simultaneously. We show that these are the only restrictions on these two dynamical invariants.

We construct an example of a fixed point free Anosov diffeomorphism of the plane, which is not topologically conjugate to a translation.

We show that various classes of products of manifolds do not support transitive Anosov diffeomorphisms. Exploiting the Ruelle–Sullivan cohomology class, we prove that the product of a negatively curved manifold with a rational homology sphere does not support transitive Anosov diffeomorphisms. We extend this result to products of finitely many negatively curved manifolds of dimension at least three with a rational homology sphere that has vanishing simplicial volume. As an application of this study, we obtain new examples of manifolds that do not support transitive Anosov diffeomorphisms, including certain manifolds with non-trivial higher homotopy groups and certain products of aspherical manifolds.