Domination between different products and finiteness of associated semi-norms

In this note we determine all possible dominations between different products of manifolds, when none of the factors of the codomain is dominated by products. As a consequence, we determine the finiteness of every product-associated functorial semi-norm on the fundamental classes of the aforementioned products. These results give partial answers to questions of M. Gromov.Comment: 7 pages; v2: small changes, to appear in Publicacions Matem\`atique

Anosov diffeomorphisms of products I. Negative curvature and rational homology spheres

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 dimensions 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.Comment: 16 pages; v2: minor changes, to appear in Ergodic Theory and Dynamical System

On a problem of Hopf for circle bundles over aspherical manifolds with hyperbolic fundamental groups

We prove that a circle bundle over a closed oriented aspherical manifold with hyperbolic fundamental group admits a self-map of absolute degree greater than one if and only if it is the trivial bundle. This generalizes in every dimension the case of circle bundles over hyperbolic surfaces, for which the result was known by the work of Brooks and Goldman on the Seifert volume. As a consequence, we verify the following strong version of a problem of Hopf for the above class of manifolds: Every self-map of non-zero degree of a circle bundle over a closed oriented aspherical manifold with hyperbolic fundamental group is either homotopic to a homeomorphism or homotopic to a non-trivial covering and the bundle is trivial. As another application, we derive the first examples of non-vanishing numerical invariants that are monotone with respect to the mapping degree on non-trivial circle bundles over aspherical manifolds with hyperbolic fundamental groups of any dimension. Moreover, we obtain the first examples of manifolds (given by the aforementioned bundles with torsion Euler class) which do not admit self-maps of absolute degree greater than one, but admit maps of infinitely many different degrees from other manifolds.Comment: 12 pages; v2: results extended to circle bundles over aspherical manifolds with hyperbolic fundamental groups, title changed accordingl

Universal partial sums of Taylor series as functions of the centre of expansion

V. Nestoridis conjectured that if $\Omega$ is a simply connected subset of $\mathbb{C}$ that does not contain $0$ and $S(\Omega)$ is the set of all functions $f\in \mathcal{H}(\Omega)$ with the property that the set $\left\{T_N(f)(z)\coloneqq\sum_{n=0}^N\dfrac{f^{(n)}(z)}{n!} (-z)^n : N = 0,1,2,\dots \right\}$ is dense in $\mathcal{H}(\Omega)$, then $S(\Omega)$ is a dense $G_\delta$ set in $\mathcal{H}(\Omega)$. We answer the conjecture in the affirmative in the special case where $\Omega$ is an open disc $D(z_0,r)$ that does not contain $0$.Comment: 9 page