# Rigidity and Small Divisors in Holomorphic Dynamics

### 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).

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.

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.

## People |
## ORCID iD |

Davoud Cheraghi (Principal Investigator / Fellow) |

Cheraghi D
(2017)

*Mathematics of Planet Earth*
Cheraghi D

*Typical orbits of quadratic polynomials with a neutral fixed point: non-Brjuno type*in Ann. Sci. 'Ecole Norm. Sup.
Cheraghi D
(2015)

*A proof of the Marmi-Moussa-Yoccoz conjecture for rotation numbers of high type*in Inventiones mathematicae
Cheraghi D

*Geometric complex analysis*
Cheraghi D
(2015)

*Satellite renormalization of quadratic polynomials*in Arxiv preprint server
Mycek P
(2017)

*Iterative solver approach for turbine interactions: application to wind or marine current turbine farms*in Applied Mathematical ModellingTitle | 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. |

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 |
Chemicals,Energy,Financial Services, and Management Consultancy,Healthcare,Manufacturing, including Industrial Biotechology |

URL | http://arxiv.org/abs/1509.07843 |

Description | Departmental Platform Grant |

Amount | £8,000 (GBP) |

Organisation | Imperial College London |

Sector | Academic/University |

Country | United Kingdom of Great Britain & Northern Ireland (UK) |

Start | 06/2016 |

End | 09/2016 |

Description | London Math Society Scheme 1 grants |

Amount | £1,669 (GBP) |

Funding ID | 11446 |

Organisation | London Mathematical Society |

Sector | Learned Society |

Country | United Kingdom of Great Britain & Northern Ireland (UK) |

Start | 09/2015 |

End | 12/2015 |

Description | Complex Feigenbaum phenomena of high type |

Organisation | Imperial College London |

Country | United Kingdom of Great Britain & Northern Ireland (UK) |

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 of Great Britain & Northern Ireland (UK) |

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 | Dynamics of high-type hedgehogs |

Organisation | Nanjing University (NJU) |

Department | Department of Mathematics |

Country | China, People's Republic of |

Sector | Academic/University |

PI Contribution | In collaboration with Professor Mitsuhiro Shishikura and Yang Fei (Nanjing University) we have sketched a proof of a trichotomy for the topology of the attractors at irrationally indifferent fixed points. This confirms a conjecture of M. Herman from the 1980. My role has been mostly on the analytic aspect of the problem. |

Collaborator Contribution | My collaborators has been mostly contributed to the combinatorial and topological aspect of the project. |

Impact | This will lead to a journal article of about 60 pages, 35 pages written to date. |

Start Year | 2016 |

Description | Dynamics of high-type hedgehogs |

Organisation | University of Kyoto |

Country | Japan |

Sector | Academic/University |

PI Contribution | In collaboration with Professor Mitsuhiro Shishikura and Yang Fei (Nanjing University) we have sketched a proof of a trichotomy for the topology of the attractors at irrationally indifferent fixed points. This confirms a conjecture of M. Herman from the 1980. My role has been mostly on the analytic aspect of the problem. |

Collaborator Contribution | My collaborators has been mostly contributed to the combinatorial and topological aspect of the project. |

Impact | This will lead to a journal article of about 60 pages, 35 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 of Great Britain & Northern Ireland (UK) |

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 | Siegel disks with boundaries of Hausdorff dimension two |

Organisation | Imperial College London |

Department | Department of Life Sciences |

Country | United Kingdom of Great Britain & Northern Ireland (UK) |

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 of Great Britain & Northern Ireland (UK) |

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 | 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 | 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 | 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 |