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Research
My research interests
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 in transcendental dynamics (click to expand!)*:
- Iteration of transcendental self-maps of the punctured plane \(\mathbb C^*=\mathbb C\setminus\{0\}\)
A transcendental self-map of \(\mathbb C^*\) is a holomorphic function \(f:\mathbb C^*\to\mathbb C^*\) for which both \(0\) and \(\infty\) are essential singularities. Such maps are all of the form
\(f(z)=z^n\exp(g(z)+h(1/z))\),
where \(n\in\mathbb{Z}\) and \(g,h\) are non-constant entire functions. The iteration of functions in this class was first considered by Rådström [Råd53], and later it was studied by Baker and Domínguez [BD98].
Dynamics of a function in the complex Arnol'd standard family,
which are examples of transcendental self-maps of the punctured plane (see D).
In black, the basin of attraction of a two periodic cycle in the unit circle.
My research focused on the study of the escaping set of such functions, which consists of the points that accummulate to \(0\) and/or \(\infty\) under iteration (roughly speaking, it is the complement of the black set above).
References:
- I. N. Baker and P. Domínguez, Analytic self-maps of the punctured plane, Complex Variables Theory Appl. 37 (1998), no. 1-4, 67-91.
- H. Rådström, On the iteration of analytic functions. Math. Scand. 1 (1953), 85-92.
My papers related to this topic:
- D. Martí-Pete, The escaping set of transcendental self-maps of the punctured plane . Ergodic Theory and Dynamical Systems 38 (2018), no. 2, 739-760.
- N. Fagella and D. Martí-Pete, Dynamic rays of bounded-type transcendental self-maps of the punctured plane. Discrete and Continuous Dynamical Systems - Series A 37 (2017), no. 6, 3123-3160.
- D. Martí-Pete, Escaping Fatou components of transcendental self-maps of the punctured plane. To appear in Mathematical Proceedings of the Cambridge Philosophical Society.
- V. Evdoridou, D. Martí-Pete and D. J. Sixsmith, Spiders' webs in the punctured plane.
To appear in Annales Academiae Scientiarum Fennicae Mathematica.
- V. Evdoridou, D. Martí-Pete and D. J. Sixsmith, On the connectivity of the escaping set in the punctured plane.
Preprint, arXiv:1908.07383.
- Baker domains
Let \(f\) be a transcendental entire function or a transcendental self-map 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, 563-576.
- P. Fatou, Sur l'itération des fonctions transcendantes entières, Acta Math. 47 (1926),
no. 4, 337-370.
- P. J. Rippon, Baker domains, Transcendental Dynamics and Complex Analysis, London Math. Soc. Lecture Note Ser., vol. 348, Cambridge Univ. Press, Cambridge, 2008, pp. 371-395.
My papers related to this topic:
- D. Martí-Pete, Escaping Fatou components of transcendental self-maps of the punctured plane. To appear in Mathematical Proceedings of the Cambridge Philosophical Society.
- V. Evdoridou, D. Martí-Pete and D. J. Sixsmith, Spiders' webs in the punctured plane.
To appear in Annales Academiae Scientiarum Fennicae Mathematica.
- V. Evdoridou, D. Martí-Pete and D. J. Sixsmith, On the connectivity of the escaping set in the punctured plane.
Preprint, arXiv:1908.07383.
- Wandering domains
Let \(f\) be a transcendental entire function or a transcendental self-map of \(\mathbb{C}^*\). We say that a component \(U\) of the Fatou set of \(f\) is a wandering domain if \(U\) is not eventually periodic, that is, for \(m,n\in\mathbb{N}\), \(f^m(U)\cap f^n(U)\neq \emptyset\) if and only if \(m=n\). Wandering domains can be classified into
- escaping: for all \(z\in U,\ f^n(z)\to \infty\) as \(n\to\infty\);
- bounded-orbit: for all \(z\in U\), there exists \(R>0\) such that \(|f^n(z)|<R\) for all \(n\in\mathbb{N}\);
- oscillating: for all \(z\in U\), there is a subsequence along which \(f^n(z)\) escapes, but there is another subsequence along which \(f^n(z)\) is bounded.
Baker [Bak76] gave the first example of an entire function with a wandering domain, which was an infinite product that he had studied previously in [Bak63]. Simpler (and explicit) examples were given later by Herman [Her84] by using holomorphic self-maps of \(\mathbb{C}^*\). These two examples were escaping wandering domains. The first oscillating wandering domain was constructed by Eremenko and Lyubich [EL92] using approximation theory.
Recently, Bishop [Bis15] constructed an example of a transcendental entire function in the so-called Eremenko-Lyubich (which have a bounded set of singular values) by using a revolutionary new technique known as quasiconformal folding. Together with M. Shishikura, we constructed another function in the same class but with finite order and the construction of which uses only quasiconformal surgery [MS19].
A sketch of the dynamics of the oscillating wandering domain
constructed in [MS19] by using quasiconformal surgery.
References:
- I. N. Baker, Multiply connected domains of normality in iteration theory, Math. Z. 81 (1963), 206-214.
- I. N. Baker, An entire function which has wandering domains, J. Austral. Math. Soc. Ser. A 22 (1976), no. 2, 173-176.
- C. J. Bishop, Constructing entire functions by quasiconformal folding, Acta Math.
214 (2015), no. 1, 1-60.
- A. E. Eremenko and M. Yu. Lyubich, Examples of entire functions with pathological
dynamics, J. London Math. Soc. (2) 36 (1987), no. 3, 458-468.
- M. Herman, Exemples de fractions rationnelles ayant une orbite dense sur la sphére
de Riemann, Bull. Soc. Math. France 112 (1984), 93-142.
My papers related to this topic:
- D. Martí-Pete, Escaping Fatou components of transcendental self-maps of the punctured plane. To appear in Mathematical Proceedings of the Cambridge Philosophical Society.
- D. Martí-Pete and M. Shishikura, Wandering domains for entire functions of finite order in the Eremenko-Lyubich class.
To appear in Proceedings of the London Mathematical Society.
- V. Evdoridou, D. Martí-Pete and D. J. Sixsmith, On the connectivity of the escaping set in the punctured plane.
Preprint, arXiv:1908.07383.
- The complex Arnol'd standard family
Arnol'd [Arn61] studied the family of analytic self-maps of the circle given by
\(S_{\alpha,\beta}(\theta)=\theta+\alpha+\beta\sin\theta\)
which he called the standard family. His motivation was to understand the dependence of the rotation number on the parameters. In this paper he introduced the so-called Arnol'd tongues, which are the sets of (real) parameters \((\alpha,\beta)\) for which the rotation number of \(S_{\alpha,\beta}\) is constant, and proved that their boundaries are analytic curves.
In her PhD, Fagella [Fag95,Fag99] studied the iteration of the complex extension of these maps, which are a family of transcendental self-maps of \(\mathbb{C}^*\) (see A). In our work in progress with M. Shishikura [MS19], we study the \(\alpha\)-parameter space of this family, where \(\alpha\in\mathbb{C}\) and \(0<\beta<1\) is fixed. We observe some finger-like structures in the lower half-plane that increase in number as \(\beta\to 0\), and we analyse them by using the theory of parabolic bifurcation.
Fingers in the \(\alpha\)-parameter space of the complex Arnol'd standard family.
References:
- V. I. Arnol'd, Small denominators I: Mapping the circle onto itself, Izv. Akad. Nauk SSSR Ser. Mat. 25 (1961), 21-86, translated in Translations of the Amer. Math. Soc. (2) 46 (1965), 213-284.
- N. Fagella, Limiting dynamics for the complex standard family, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 5 (1995), no. 3, 673-699.
- N. Fagella, Dynamics of the complex standard family, J. Math. Anal. Appl. 229 (1999), no. 1, 1-31.
My papers related to this topic:
- D. Martí-Pete and M. Shishikura, Fingers in the parameter space of the complex Arnol'd standard family.
In preparation.
* Note that several of my papers are related to more than one of my research interests.
My supervisors and host professors
Research groups
Research visits
I visited / will visit
- Prof. Mistuhiro Shishikura at Kyoto University (Japan), December 6-22, 2014
- Dr. Vasiliki Evdoridou at The Open University (UK), March 2-9, 2019
- Dr. Vasiliki Evdoridou at The Open University (UK), September 3-9, 2019
- Dr. Anna Miriam Benini at Universita di Parma (Italy), November, 2019
I hosted / will host
- Dr. Anna Miriam Benini (Universitat de Barcelona) at Kyoto University, November 3-11, 2017
- Dr. Vasiliki Evdoridou (The Open University) at IMPAN, May 12-18, 2019
- Prof. Mistuhiro Shishikura (Kyoto University) at IMPAN, August 12-22, 2019
- Dr. Kirill Lazebnik (Caltech) at IMPAN, September 22-24, 2019
- Dr. Matthew Jacques (The Open University) at IMPAN, October 21-23, 2019
Funding
I am very thankful for the financial support from
- Spanish Ministery of Science, Innovation and Universities grant "Sistemas Dinámicos Holomorfos" MTM2011-26995-C02-02 (PI: Prof. Núria Fagella)
- Research budget for PhD students by The Open University (3 x 1,250 GBP/year)
- Santander Formula Scholarship 2014 (5,000 EUR)
- Postgraduate Research Conference Grants (Scheme 8) by the London Mathematical Society (3,969.40 GBP) (which were complemented by an extra 1,500 GBP by The Open University)
- Japan Society for the Promotion of Science grant-in-aid 16F16807 (2,100,000 JPY)
- IMPAN Young Researchers grants, winter edition 2019 (5,000 PLN)
- National Science Centre, Poland, grant 2016/23/P/ST1/04088 (PI: Dr. Artem Dudko) under the POLONEZ programme which has received funding from the EU Horizon 2020 research and innovation programme under the MSCA grant agreement No. 665778 (see info at IMPAN)
as well as the times I have been supported by conferences or other researchers' grants.
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.
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