
Research
My research area is Complex Dynamics, which lies in between Dynamical Systems and Complex Analysis. We study the discrete dynamical systems associated to the iteration of holomorphic maps.
More precisely, I am interested in the following topics (click to expand!):
 iteration of holomorphic selfmaps of the punctured plane \(\mathbb C^*=\mathbb C\setminus\{0\}\)
Under construction...
 the escaping set of transcendental functions
Under construction...
 Baker domains
Let \(f\) be a transcendental entire function or a transcendental selfmap of \(\mathbb{C}^*\). We say that a component \(U\) of the Fatou set of \(f\) is a Baker domain if \(U\) is periodic, that is, there is \(p\in\mathbb{N}\) such that \(f^p(U)\cap U\neq \emptyset\), and moreover \(U\) is contained in the escaping set \(I(f)\).
Fatou [Fat26] gave the first example of a function with a Baker domain, \(F(z)=z+1+e^{z}\) (see picture below). For transcendental entire functions, Baker [Bak84] proved that Baker domains are simply connected (in general, this is not the case for meromorphic functions). See [Rip08] for a survey on Baker
domains.
In gray, the Baker domain of the function \(F(z)=z+1+e^{z}\).
This picture was used in the paper Fatou's web by V. Evdoridou and was generated using software provided by A. Chéritat.
References:
 I. N. Baker, Wandering domains in the iteration of entire functions. Proc. London Math. Soc. 49 (1984), no. 3, 563576.
 P. Fatou, Sur l'itération des fonctions transcendantes entières, Acta Math. 47 (1926),
no. 4, 337370.
 P. J. Rippon, Baker domains, Transcendental Dynamics and Complex Analysis, London Math. Soc. Lecture Note Ser., vol. 348, Cambridge Univ. Press, Cambridge, 2008, pp. 371395.
My papers related to this topic:
 D. MartíPete, Escaping Fatou components of transcendental selfmaps of the punctured plane. To appear in Mathematical Proceedings of the Cambridge Philosophical Society.
 V. Evdoridou, D. MartíPete and D. J. Sixsmith, Spiders' webs in the punctured plane.
Preprint, arXiv:1901.05276.
 wandering domains
Under construction...
 the complex Arnol'd standard family
Under construction...
 iteration of holomorphic functions with several essential singularities
Under construction...
Analysis and geometry group at OU: www.mathematics.open.ac.uk/research/pure/analysisandgeometry
Holomorphic dynamics group at UB: www.maia.ub.es/holodyn
Dynamical systems group at KU: https://www.math.kyotou.ac.jp/dynamics/
My page at the GSDUAB: www.gsd.uab.cat/personal/dmarti
My supervisors and host professors
The Complex Dynamics group in Milton Keynes
From left to right: Phil Rippon, Gwyneth Stallard, Dave Sixsmith, Vasso Evdoridou and me. This picture was taken in front of the Pisa tower in October 2013 – the sun dazzled!
The Complex Dynamics group in Barcelona
From left to right: Sébastien Godillon, Xavier Jarque, Núria Fagella, me, Jordi Canela and Toni Garijo. This picture was taken in the Annual Meeting 2013 in Tarragona.
The whole family! In the front row, the advisors: Núria Fagella and Xavier Jarque. In the back row, the students: Toni Garijo, Asli Deniz, Jordi Canela, Jordi Taixés and me. This is from one of our Complex Dynamics Working Seminar in January 2013 at the IMUB Room in Barcelona.



