Rigidity and Small Divisors in Holomorphic Dynamics

Lead Research Organisation: Imperial College London
Department Name: Dept of Mathematics

Abstract

The simplest non-linear systems are driven by quadratic polynomials. That is, "time n" of a state is determined by a quadratic polynomial of "time n-1" of that state. However, despite over a century of intense study, the dynamical features of even quadratic formulae remain far from well understood. For example, complex quadratic polynomials with "small divisors", which may be used to model resonance phenomena, still exhibit mysterious behaviour in many cases.

There has been extensive research on the dynamics of quadratic polynomials over the last three decades. Often, sophisticated tools from different disciplines of mathematics are needed to describe the fine dynamical features of these maps. Usually, a set of such tools is introduced to study the dynamics of a type of quadratic maps, but leads to the successful study of non-linear systems of that type. Thus, an effective set of tools for the study of quadratic polynomials provide the basis of extensive research in the wider area of non-linear systems.

In this project, I develop a new set of tools from different disciplines of mathematics to provide a comprehensive description of the dynamics of certain types of quadratic polynomials. This develops effective techniques from analysis, geometry, and more sophisticated mathematical machinery such as renormalisation and Teichmuller theory.
I will achieve the following major goals.

(1) Small divisors:
A main goal of this research is to introduce a systematic approach to obtain a comprehensive understanding of the dynamics of quadratic polynomials with small divisors. This provides the first examples of such systems with unstable behavior at the center of resonance, whose dynamical behaviour is completely understood.

The Julia set of a quadratic polynomial is the unstable locus of its dynamics. A recent remarkable result of X. Buff and A. Cheritat states that there are quadratic polynomials with small divisors which have observable (positive area) Julia sets. A central problem in the presence of small divisors is to determine arithmetic conditions on the rotation number that leads to observable Julia sets. The proposed research makes major advances on this problem.

(2) Rigidity and density of Hyperbolicity:
The quadratic polynomials that exhibit a certain well understood dynamical behaviour are called hyperbolic. There is a remarkable property, anticipated by P. Fatou in 1920's, stating that any quadratic polynomial may be perturbed to a nearby one with hyperbolic behaviour (by small changes in coefficients in an appropriate normalisation).

The project studies some deep analytic properties of a renormalisation technique to confirm this conjecture for certain types of quadratic polynomials (a Cantor set of parameters). This programme suggests a refined quantitative (in spirit of continued fractions) version of this conjecture to hold.

(3) Generalized Feigenbaum maps:
Period doubling bifurcation is a remarkable phenomenon that appears in the family of quadratic polynomials with real coefficients. There is a wide range of analogous, but more complicated, phenomena that occur when one considers quadratic polynomials with complex coefficients. This reflects the complicated structure of the Mandelbrot set. The dynamical features of such maps with real coefficients have been deeply studied in a period of intense research in 1980's and 90's, while the ones with complex coefficients are largely unexplored. The research proposal uses renormalisation techniques and develops innovative analytical methods to present a detailed description of the dynamics of such a map near degenerate bifurcations.

I will carry out some parts of this major project in collaboration with the leading experts of holomorphic dynamics: A. Avila (Rio, Brazil and Paris, France), X. Buff (Toulouse, France), A. Cheritat (Bordeaux, France), and M. Shishikura (Kyoto, Japan).

Planned Impact

The field of Dynamical Systems plays a central role in the development of Mathematics and Physics. It is widely applied in many disciplines to study long term behaviour of environmental, economic, and social systems, such as predicting average values of observables over long periods of time. Thus, the impacts of advances in dynamical systems on everyday life is very important.

The proposed project concerns foundational work to make breakthroughs on central conjectures of Dynamical Systems. It introduces and develops techniques from different disciplines of mathematics such as analysis, geometry, and Diophantine approximation, to provide a comprehensive understanding of highly complicated dynamical behaviours. The flexibility of the methods developed in this programme are very likely to be utilised in a wide range of applications. In particular, the proposed research impacts the areas listed below.

(1) The techniques developed in this research can be used to describe the dynamics of systems with resonances. Resonances are prevalent phenomena in almost periodic events, from the rising of the sun each day to more complicated electromagnetic waves. They often lead to mysterious behaviours. Many such systems, even when given by simple formulae like quadratic polynomials, have remain far from understood to date. One of the main aims of this project is to introduce a systematic approach to successfully study such systems modeled by quadratic formulae on the complex plane. The project answers questions such as whether the set of unstable states are observable (have non-zero probability of occurring).

(2)The project introduces cost effective algorithms for simulating highly complicated non-linear systems. These are dynamical systems arising from complicated bifurcation patterns. Through the developments of this project, I plan to develop software for simulating such systems and making them widely accessible through the internet. I plan to deliver lectures addressing the general public to share the excitements of these ideas and the challenges involved.

(3)The project introduces effective methods that immediately impact many areas such as shape analysis and medical imaging (in healthcare industries), electrical impedance tomography, analysis of water waves. One of the main building blocks of the proposed project is to develop effective analytic methods to describe fine geometric features of the solutions of non-linear partial differential equations, and to obtain optimal estimates on the dependence of the solution of such equations on the data. These methods can be used to establish estimates on conformal mappings and in conformal geometry which have found wide ranges of applications listed above.

Publications

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Avila A (2018) Statistical properties of quadratic polynomials with a neutral fixed point in Journal of the European Mathematical Society

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Cheraghi D (2017) Mathematics of Planet Earth

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Cheraghi D (2015) Satellite renormalization of quadratic polynomials in Arxiv preprint server

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Cheraghi D (2017) Topology of Irrationally indifferent attractors in Arxiv Preprint server

 
Title Siegel disks 
Description Maximal linearisation domains of non-linear systems have been produced. It involves writing intelligent software codes that simulated some time -consuming tasks in shorter periods of times. The current programme requires 5 full-working days for a normal computer to obtain a single image. 
Type Of Art Image 
Year Produced 2015 
Impact Helped with developing mathematical methods to study the dynamics of non-linear systems. 
URL http://wwwf.imperial.ac.uk/~dcheragh/Siegel.html
 
Description We have developed powerful mathematics methods to tackle a central problem in mathematics that was left open since 1970's. At this point we have obtained a complete topological description of long term behaviour of a fundamental system with resonant behaviour. This is collectively known as the problem of small divisors.
Exploitation Route This originates a method to study the long term behaviour of some non-linear systems that were out of reach until recently.
Sectors Aerospace, Defence and Marine,Energy,Financial Services, and Management Consultancy,Healthcare,Manufacturing, including Industrial Biotechology

URL https://arxiv.org/abs/1706.02678
 
Description Departmental Platform Grant
Amount £8,000 (GBP)
Organisation Imperial College London 
Sector Academic/University
Country United Kingdom
Start 06/2016 
End 09/2016
 
Description European Partners Fund
Amount £5,000 (GBP)
Organisation Imperial College London 
Sector Academic/University
Country United Kingdom
Start 09/2017 
End 10/2019
 
Description H2020
Amount € 183,454 (EUR)
Organisation European Research Council (ERC) 
Sector Public
Country European Union (EU)
Start 10/2018 
End 09/2020
 
Description H2020- Marie-Curie Individual fellowships
Amount € 224,933 (EUR)
Funding ID LYP-RIG - GAP-837602 
Organisation European Research Council (ERC) 
Sector Public
Country European Union (EU)
Start 07/2019 
End 07/2021
 
Description London Math Society Scheme 1 grants
Amount £1,670 (GBP)
Funding ID 11446 
Organisation London Mathematical Society 
Sector Learned Society
Country United Kingdom
Start 09/2015 
End 12/2015
 
Description London Mathematical Society Scheme 1 grants
Amount £4,000 (GBP)
Organisation London Mathematical Society 
Sector Learned Society
Country United Kingdom
Start 01/2018 
End 05/2018
 
Description Complex Feigenbaum phenomena of high type 
Organisation Imperial College London
Country United Kingdom 
Sector Academic/University 
PI Contribution Davoud Cheraghi is the PhD supervisor of the candidate (M. Pedramfar).
Collaborator Contribution The PhD student is collaborating on some aspects of the research proposal submitted to EPSRC.
Impact The collaboration has lad to a clear programme to describe the behaviour of one of the most complicated phenomena in non-linear dynamics; the presence of renormalisation structures. I expect this lead to two major papers about 50 pages each. About 15 pages of the first paper is written to date.
Start Year 2016
 
Description Computational complexity of Lorenz attractors 
Organisation Meteorological Office UK
Country United Kingdom 
Sector Public 
PI Contribution In this collaboration with Dr Gabriel Rooney we investigate the use of renormalisation methods in simulations related to weather forecasting and environmental changes in oceans. I provide expertise from dynamical systems.
Collaborator Contribution The collaborator provides expertise on applications.
Impact We have outlined a PhD project on this collaboration, which is funded by a CDT at Imperial College London.
Start Year 2017
 
Description Dimension paradox of irrationally indifferent attractors 
Organisation Imperial College London
Department Department of Mathematics
Country United Kingdom 
Sector Academic/University 
PI Contribution In collaboration with Dr Yang Fei and Alexandre de Zotti we have successfully study the metric m[properties of the attractors of analytic maps with resonant behavior. A journal article on this study is near completion. I have played a central role in drawing the overall strategy, and providing the foundational ingredients of the work.
Collaborator Contribution My collaborators have been mostly contributing to thechnical aspect of the project.
Impact This will lead to a journal article of about 40 pages, about 30 pages written to date.
Start Year 2016
 
Description Dimension paradox of irrationally indifferent attractors 
Organisation Nanjing University (NJU)
Department Department of Mathematics
Country China 
Sector Academic/University 
PI Contribution In collaboration with Dr Yang Fei and Alexandre de Zotti we have successfully study the metric m[properties of the attractors of analytic maps with resonant behavior. A journal article on this study is near completion. I have played a central role in drawing the overall strategy, and providing the foundational ingredients of the work.
Collaborator Contribution My collaborators have been mostly contributing to thechnical aspect of the project.
Impact This will lead to a journal article of about 40 pages, about 30 pages written to date.
Start Year 2016
 
Description Endomorphisms of C2 with a wandering domain tending to a Cantor set 
Organisation Imperial College London
Department Department of Mathematics
Country United Kingdom 
Sector Academic/University 
PI Contribution This is a collaboration with Professor Sebastian van Strien, DR Trevor Clark, and Fabrizio Bianchi. I have sketched the main strategy of the project where we investigate the existence of wandering domains in higher dimensional analytic spaces.
Collaborator Contribution Provide technical details from real analysis and higher dimensional complex analysis.
Impact If successful, this will lead to a journal article.
Start Year 2017
 
Description Hairy Cantor sets in the plane 
Organisation Imperial College London
Department Department of Mathematics
Country United Kingdom 
Sector Academic/University 
PI Contribution We discovered a new topological object in the plane with universal features similar to the Cantor set, that is, the set can be described by a number of axioms, and any two such objects in the plane are ambiently homeomorphic. Surprisingly, such objects which have not been identified to date, are prevalent in analytic dynamics. A dense set of rational maps on the bifurcation locus, preserves such an object.
Collaborator Contribution Technical aspects of the project have been carried out by my PhD student Mohammad Pedramfar.
Impact A journal paper about 30 pages will emerge from this work.
Start Year 2018
 
Description Siegel disks with boundaries of Hausdorff dimension two 
Organisation Imperial College London
Department Department of Life Sciences
Country United Kingdom 
Sector Academic/University 
PI Contribution In this project we investigate the existence of maximal linearisation domains with large boundaries, that is, of Heusdorff dimension two. This is a joint project with my research associate Dr Alexandre De Zotti.
Collaborator Contribution Provides technical support.
Impact The collaboration will lead to a journal paper.
Start Year 2016
 
Description Topology of isentropes in a two parameter family of unimodal maps 
Organisation Imperial College London
Department Department of Mathematics
Country United Kingdom 
Sector Academic/University 
PI Contribution This is project with my colleague at Imperial college London where we study the global deformation structures in the parameter space of polynomials. I provide techniques from complex analysis.
Collaborator Contribution The partner provides techniques from real dynamics.
Impact Expect to finish a paper about 20 pages, of which 10 pages written down to date.
Start Year 2016
 
Description Universality of the scaling laws 
Organisation University of Kyoto
Department Department of Mathematics
Country Japan 
Sector Academic/University 
PI Contribution In this joint collaboration between PI(Cheraghi) and Professor Mitsuhiro Shishikura, we have made a breakthrough on a conjecture of physicists from the 1970's on the "universality of the scaling laws" in generic families of analytic transformations. This has been a collaborative research carried out over several years, and only completed in September 2015. It is not possible to draw a line between the role of the PI and the partner.
Collaborator Contribution In this joint collaboration between PI(Cheraghi) and Professor Mitsuhiro Shishikura, we have made a breakthrough on a conjecture of physicists from the 1970's on the "universality of the scaling laws" in generic families of analytic transformations. This has been a collaborative research carried out over several years, and only completed in September 2015. It is not possible to draw a line between the role of the PI and the partner.
Impact A preprint of this article (73 pages) is now available on the Arxiv preprint server.
Start Year 2016
 
Description simultanious linearisation of commuting germs tangent to rotations 
Organisation Pierre and Marie Curie University - Paris 6
Country France 
Sector Academic/University 
PI Contribution expertise in iterations of complex analytic maps of the complex plane
Collaborator Contribution expertise in iterations of real analytic maps of the circle
Impact expect a journal paper resulting from this collaboration
Start Year 2017
 
Description Five lectures on dynamical systems 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach National
Primary Audience Postgraduate students
Results and Impact 15 postgraduate students attended five lectures of two hours each. This was part of a Centre for doctoral training in mathematics of planet earth.
Year(s) Of Engagement Activity 2015
 
Description Five lectures on dynamical systems for Mathematics of Planet Earth 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach National
Primary Audience Postgraduate students
Results and Impact Delivered five lectures (of two hours each) to nonspecialists in Mathematics of Planet Earth
Year(s) Of Engagement Activity 2016
 
Description Junior Analysis seminars 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach National
Primary Audience Postgraduate students
Results and Impact I initiated the Junior Analysis seminars at Imperial College, in order to bring the PHD students working in broader analysis area together in order to broaden their scope, practice communicating their work, and identify phd students at other institutions within the UK. The meeting occurs every Friday for two hours, 2-3, and 3:30 to 4:30.
Year(s) Of Engagement Activity 2016,2019
URL https://www.imperial.ac.uk/pure-analysis-and-pdes/seminars/jas/
 
Description London Analysis and Probability seminar 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach National
Primary Audience Professional Practitioners
Results and Impact I was one of the organisers of the regular biweekly seminars on broader analysis are in the London region. This is a joint seminar between Imperial College London, University College London, Kings College London, and Queen Mary University London. I am the contact point for Imperial college London.
Year(s) Of Engagement Activity 2017,2018
 
Description MPE CDT Sandpit meeting 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach National
Primary Audience Industry/Business
Results and Impact In this meeting experts from different areas of natural sciences came together to discuss techniques that could be useful for the study of environmental problems. Most of the participants came from industry and national organisations like MET Office Thames Water. My discussion with some of the participants from MET Office has lead to a proposal for a PhD thesis in CDT on Mathematics of Planet Earth. In the meeting I presented a brief description of how renormalisation methods could be used in more efficient programming methods in whether forecasting.
Year(s) Of Engagement Activity 2016
 
Description Organisation of a conference 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Postgraduate students
Results and Impact This was an international conference to bring Scandinavian experts in analysis in contact with dynamics experts in the UK. The meeting was highly successful in exchanging key challenges and the required tools, and due to high level of interest from both sides, it is planned that the meeting will take place every other year, alternating between UK and Finland.
Year(s) Of Engagement Activity 2018
URL https://personalpages.manchester.ac.uk/staff/tuomas.sahlsten/analysisdynamics/
 
Description Parameter problems in analytic dynamics 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Postgraduate students
Results and Impact This was a major international conference on analytic dynamics which brought together the leading experts in the field. The event has helped us with attracting the best in the field and some of the best in the research area wish to visit our group at later stages. For instance, we have already made applications for Professor Francois Berteloot (University of Toulouse, France) to spend six months at Imperial College, and Professor Genadi Levin (University of Jerusalem, Israel) has requested to spend a six month sabbatical at Imperial College. We have also had a number of very strong applicants for junior level positions at Imperial College.
Year(s) Of Engagement Activity 2016
URL http://wwwf.imperial.ac.uk/~dcheragh/PPAD/Conference.html
 
Description Pure Analysis and PDE 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Postgraduate students
Results and Impact I was the organiser of the regular weekly research seminars on Pure Analysis and PDE in the department of mathematics.
Year(s) Of Engagement Activity 2017