On Closed Invariant Sets in Local Dynamics

We investigate the dynamical behaviour of a holomorphic map on a $f-$invariant subset $\mathcal{C}$ of $U,$ where $f:U \to \mathbb{C}^k.$ We study two cases: when $U$ is an open, connected and polynomially convex subset of $\mathbb{C}^k$ and $\mathcal{C} \subset \subset U,$ closed in $U,$ and when $\partial U$ has a p.s.h. barrier at each of its points and $\mathcal{C}$ is not relatively compact in $U.$ In the second part of the paper, we prove a Birkhoff's type Theorem for holomorphic maps in several complex variables, i.e. given an injective holomorphic map $f,$ defined in a neighborhood of $\overline{U},$ with $U$ star-shaped and $f(U)$ a Runge domain, we prove the existence of a unique, forward invariant, maximal, compact and connected subset of $\overline{U}$ which touches $\partial U.$Comment: Exposition has been improved; Corollary 3.6 has been corrected; 8 pages; version close to be publishe

M\"obius transformations and the Poincar\'e distance in the quaternionic setting

In the space \hh of quaternions, we investigate the natural, invariant geometry of the open, unit disc \Delta_{\hh} and of the open half-space \hh^{+}. These two domains are diffeomorphic via a Cayley-type transformation. We first study the geometrical structure of the groups of M\"obius transformations of \Delta_{\hh} and \hh^{+} and identify original ways of representing them in terms of two (isomorphic) groups of matrices with quaternionic entries. We then define the cross-ratio of four quaternions, prove that, when real, it is invariant under the action of the M\"obius transformations, and use it to define the analogous of the Poincar\'e distances on \Delta_{\hh} and \hh^{+}. We easily deduce that there exists no isometry between the quaternionic Poincar\'e distance of \Delta_{\hh} and the Kobayashi distance inherited by \Delta_{\hh} as a domain of $\mathbb{C}^{2}$, in accordance with a direct consequence of the classification of the non compact, rank 1, symmetric spaces.Comment: Details added in proof of Theorem 6.4. Exposition improved in some ambiguous points. References added. 28 page

Log-biharmonicity and a Jensen formula in the space of quaternions

Given a complex meromorphic function, it is well defined its Riesz measure in terms of the laplacian of the logarithm of its modulus. Moreover, related to this tool, it is possible to prove the celebrated Jensen formula. In the present paper, using among the other things the fundamental solution for the bilaplacian, we introduce a possible generalization of these two concepts in the space of quaternions, obtaining new interesting Riesz measures and global (i.e. four dimensional), Jensen formulas.Comment: Final Version. To appear on Annales Academiae Scientiarum Fennicae Mathematica, Volume 44 (2019

A remark on the Ueno-Campana's threefold

We show that the Ueno-Campana's threefold cannot be obtained as the blow-up of any smooth threefold along a smooth centre, answering negatively a question raised by Oguiso and Truong.Comment: To appear on Michigan Math. Journal, Vol. 65 (2016

Localized intersection of currents and the Lefschetz coincidence point theorem

We introduce the notion of a Thom class of a current and define the localized intersection of currents. In particular we consider the situation where we have a smooth map of manifolds and study localized intersections of the source manifold and currents on the target manifold. We then obtain a residue theorem on the source manifold and give explicit formulas for the residues in some cases. These are applied to the problem of coincidence points of two maps. We define the global and local coincidence homology classes and indices. A representation of the Thom class of the graph as a Cech-de~Rham cocycle immediately gives us an explicit expression of the index at an isolated coincidence point, which in turn gives explicit coincidence classes in some non-isolated components. Combining these, we have a general coincidence point theorem including the one by S. Lefschetz

The harmonicity of slice regular functions

In this article we investigate harmonicity, Laplacians, mean value theorems and related topics in the context of quaternionic analysis. We observe that a Mean Value Formula for slice regular functions holds true and it is a consequence of the well known Representation Formula for slice regular functions over $\mathbb{H}$. Motivated by this observation, we have constructed three order-two differential operators in the kernel of which slice regular functions are, answering positively to the question: is a slice regular function over $\mathbb{H}$ (analogous to an holomorphic function over $\mathbb{C}$) "harmonic" in some sense, i.e. is it in the kernel of some order-two differential operator over $\mathbb{H}$ ? Finally, some applications are deduced, such as a Poisson Formula for slice regular functions over $\mathbb{H}$ and a Jensen's Formula for semi-regular ones.Comment: The exposition of this paper has been improved a lot following the valuable suggestions of a careful Referee that we warmly thank. The paper will appear soon on The Journal of Geometric Analysi

On a quaternionic Picard theorem

The classical theorem of Picard states that a non-constant holomorphic function $f:\mathbb{C}\to\mathbb{C}$ can avoid at most one value. We investigate how many values a non-constant slice regular function of a quaternionic variable $f:\mathbb{H}\to\mathbb{H}$ may avoid.Comment: 15 pages. To appear on Proc. Americ. Math. Soc. (2020

On Brolin's theorem over the quaternions

In this paper we investigate the Brolin's theorem over $\mathbb{H}$, the skew field of quaternions. Moreover, considering a quaternionic polynomial $p$ with real coefficients, we focus on the properties of its equilibrium measure, among the others, the mixing property and the Lyapunov exponents of the measure. We prove a central limit theorem and we compute the topological entropy and measurable entropy with respect to the quaternionic equilibrium measure. We prove that they are equal considering both a quaternionic polynomial with real coefficients and a polynomial with coefficients in a slice but not all real. Brolin's theorems for the one slice preserving polynomials and for generic polynomials are also proved.Comment: 27 pages. To appear on Indiana University Mathematics Journal (2021). We are really in debt to the anonymous referee for having read so carefully our paper, letting us to improve a lot its expositio