Sixth Framework Programme

Marie Curie Actions

 

EU Research Training Network CODY

Conformal Structures and Dynamics

Research Training Networks
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Project

Milestones Objectives Workpackages

Workpackages

The research is split into workpackages and tasks to address the objectives. The following table shows the timetable of these work packages. Below you find the descriptions of the specific work packages and tasks.

table

Work Package I: Conformal structures, analytic and geometric background and view; Potential theory, analytic tools (A1-A2)

This workpackage is one of the ‘engines’ of the proposal and includes for example the aim to generalize classical theory about quasiconformal mappings to more general settings, applications of holomorphic motions, removability of sets.

Leader: Coordinator for this workpackage is partner 3 [UH].

Task I.1. To study mappings with unbounded but controlled quasiconformal dilatation.
·1 Partner 3 [UH]: Prove results which are relevant for quasiconformal surgery of complex dynamical systems; for example in relation to boundaries of hyperbolic components where one needs results about mappings with unbounded but controlled distortion.
·2 Collaboration through secondments will be necessary.
Milestones: M I.1.
Addresses: Research Objective A1-A2.

Task I.2. To study mappings of finite distortion with random coefficients.
·1 Partner 3 [UH]: Optimal regularity and removability results for mappings with exponentially decreasing dilatation; rigidity estimates for mappings of finite distortion.
·2 Collaboration through secondments will be necessary.
Milestones: M I.2
Addresses: Research Objective A1-A2.

Task I.3. Singular integrals on general settings including metric measure spaces.
·1 Partner 3 [UH]: Establish conditions under which boundedness or convergence imply rectifiability (geometric regularity) of the underlying measure.
·2 Collaboration through secondments will be necessary.
Milestones: M I.2
Addresses: Research Objective A1-A2.

Work Package II : Topics in Fractals and Multifractal Analysis (A3, see also A2, P1-2† and themes D)†

This workpackage includes to work on a general theory for computing fractal dimensions and multifractal description of various measures associated to dynamical systems.

Leader: Coordinators for this workpackage are partners 2 [IMPAN] and 8 [TEI].

Task II.1. Hausdorff dimension and multi-fractal analysis of singular measures.
·1 Partner 8 [TEI]: Calculate the Hausdorff dimension of several sets characterized by their digits in some base. Study the Salem sets and more generally the relation of the Hausdorff dimension of a measure with respect to its Fourier transform and several asymptotic relations.
·2 The maximal benefit from the expertise in the network will be reached by appointing a number of ER and ESR’s and by secondments with other teams.
Milestones: M II.1
Addresses: Research Objective A3.

Task II.2. Building multiscale transforms based on wavelet–type techniques on fractal spaces generated by anomalous scaling laws.
·1 Partner 8 [TEI]: Detect data emerging from IFS approximation, with a view to explore the scaling limits of the cascade approximations and the rate of approximation spaces
·2 The maximal benefit from the expertise in the network will be reached by appointing a number of ER and ESR’s and by secondments with other teams.
Milestones: M II.1
Addresses: Research Objective A3.

Task II.3. Strange attractors and thermodynamical formalism.
Partner 2 [IMPAN]: The aim is to make progress in constructing a general theory, involving projection and intersection schemes in relation with geometric measure theory (in relation with A1, A2) based on recent breakthroughs, in dimension 1 and higher (conformal and non-conformal). Applications in geophysical fluid dynamics, on porous medium flow, biological systems (and – if possible - also on multifractal analysis of electrocardiograms) will be made.
Milestones: M II.1
Addresses: Research Objective A3.

Work Package III : Iteration of interval and circle maps, and the complexification, weak hyperbolicity and physical measures (D1) and Beyond Dimension One (D5)

This workpackage includes also the applications of quasiconformal surgery, holomorphic motions and to deepen the understanding of holomorphic dynamical systems.
Leader: Coordinators for this workpackage are partners 1 [Warwick] and 4 [CNRS].

Task III.1. Physical measures.
·1 Partner 1 [Warwick]: The purpose of this project is to consider families of real polynomials, and prove that within such families for Lebesgue almost all parameters the systems have a physical measure. The main problem here is to transfer information in state space to information in parameter space. In the case when one has one critical point there are well-established techniques for doing this (based on the lambda-lemma), but in the case where one has several critical points new ideas will need to be developed.
·2 Regular contact, and possibly a secondment are envisaged.
Milestones: M III.2
Addresses: Research Objective D1.

Task III.2. Size of tongues.
·1 Partner 4 [CNRS]: Given a family f_t of circle diffeomorphisms with rotation number rho(t). Determine the size of the tongues associated to rational values of rho(t). The purpose of this project is to generalize the results obtained by Cheritat concerning the asymptotic tail of the analytic parabolic germ through the small divisor approach. This task will be performed by a PhD student to be appointed as ESR. Regular contact with others in France, in particular with Xavier Buff will be crucial for this project.
Milestones: M III.2
Addresses: Research Objective D1 and D2.

Task III.3. Higher dimension.
·2 Partner 1 [Warwick]: To make some progress on a no-wandering domain theorem in dimension two. This is an extremely challenging task.
·3 Partner 4 [CNRS]: To make further progress in understanding Henon maps in C^2.
Milestones: M III.1
Addresses: Research Objective D1 and D5.

Work Package IV : Geometry of dynamical and parameter space (D2)

Leader: Coordinators for this workpackage are partners 4 [CNRS] and 7 [RUC].

Task IV.1. Yoccoz puzzle and parapuzzles in the multicritical case.
·1 Partner 7 [RUC]: Obtain combinatorial and analytical/geometric descriptions of the cubic connectedness locus and corresponding dynamics.
·2 Regular contact, and possibly a secondment, with the French and possibly UK team.
Milestones: M IV.2
Addresses: Research Objective D2.

Task IV.2. Newton's method for entire functions.
·0 Partner 6 [CAU]: To study Newton's method for general classes of entire functions. In particular, find "small" sets of starting values that find all roots and to show that the area (in the plane) of the attracting basins is infinite for functions of small order of growth.
·1 There will be regular contact in particular with Bremen and possibly secondment with work on related projects in partner 5 [UB].
Milestones: M IV.3.
Addresses: Research Objective D2.

Task IV.3. Bifurcation of parabolic points.
·0 Partner 4 [CNRS]: Describe bifurcations of parabolic points through approximations of holomorphic vector field
·1 Close contact with various people in France, in particular with Tan Lei. Collaboration with partner 1 [Warwick].
Milestones: M IV.1
Addresses: Research Objective D2.

Task IV.4. Geometric limits.
·0 Partners 4 [CNRS] and 1 [Warwick]: Study geometric limits, parabolic renormalizations and iterations of transcendental functions.
Milestones: M IV.1
Addresses: Research Objective: D2.

Task IV.5. Study the geometry of dynamical and parameter spaces of families of entire (or meromorphic) transcendental functions.
·0 Partner 5 [UB]: The geometry of dynamical and parameter spaces of families of entire (or meromorphic) transcendental functions.
·1 Regular contact with other teams, and possibly secondment, will be essential.
Milestones: M V.2
Addresses: Research Objective D2.

Work Package V : Hausdorff measure and dimension. Limit sets of Kleinian groups and relations. Iterated Function Systems. (D3,D4,D6)

This workpackage includes questions about Lebesgue measure and conformal measures for Julia sets. In spite of recent progress, much remains unknown about the dimension, measure and conformal measures of Julia sets and Kleinian groups.

Leader: Coordinators for this workpackage are partners 6 [CAU] and 2 [IMPAN].

Task V.1. Hausdorff and Lebesgue measure of Julia sets.
·0 Partner 6 [CAU]: Obtain results about Hausdorff dimension and area of Julia sets for certain classes of functions with unbounded set of singularities. Other conditions that play a role here are suitable notions of hyperbolicity. Questions concerning the area of Julia sets are closely connected to ergodicity, and such problems are also part of the research task.
·1 There will be regular contact in particular with Bremen and Goettingen and with other teams.
Milestones: M V.1
Addresses: Research Objective D3-D4.

Task V.2. Determine conformal measures for various classes of Julia sets.
·0 Partners 2 [IMPAN] and 4 [CNRS]: develop the method of inducing leading to infinite IFS and to investigate periodic orbits and the dynamical Zeta-function. Obtain deeper understanding of the hairy structure of Julia sets for entire/meromorphic functions in general; more comprehensive theory fat Julia set; results on the continuity of Hausdorff dimension of Julia sets and study of the problem of the continuity for Julia-Lavaurs sets.
Milestones: M V.2 and M V.3
Addresses: Research Objective D3-D4.

Task V.3. Iterated Function Systems (IFS) (D6)
·0 Partner 2 [IMPAN]: Improve the understanding of IFS with overlaps (Hausdorff and packing measures of limit sets). Determine the dimension of sets of parameters with defected dimension of the limit set and study probability distribution in the "fat" case (starting from recent breakthroughs by Avila, Tsujii, Gouezel and others). (Non)uniqueness of dimension equilibria will be related to work by Rams.
Milestones: M V.4
Addresses: Research Objective D6.

Work Package VI : Scaling limits, conformal invariance and universality. Infinite dimensional systems and Turbulent transport (P1-P3).

Leader: This workpackage will be coordinated by partners 9 [UNIGE] and 4 [CNRS].

Task VI.1. Conformal invariance.It is widely believed that many 2D lattice models of physical phenomena (percolation, Ising model, self-avoiding polymers, ...) have conformally invariant and universal scaling limits, which enabled physicists to make striking predictions about exact values of dimensions and scaling exponents. Recently mathematicians greatly advanced in understanding these conjectures. Much of the progress was based on the new process introduced by Schramm, the Stochastic Loewner Evolution (SLE), which gives all conformally invariant curves which can arise as scaling limits. This leads to the following projects:
·1 Partner 9 [UNIGE]: Study the properties of SLEs:.dimensional properties of random sets, and properties of SLEs suggested by analogy with interfaces in lattice models.
·2 Partner 9 [UNIGE]: Construct scaling limits for lattice models. Investigate Ising interfaces on Riemann surfaces and understand universality (i.e. independence of the limit of the lattice).
Milestones: M VI.1
Addresses: Research Objective P1.

Task VI.2. Develop foundations for renormalisation.
·0 Partners 9 [UNIGE] and 3 [UH]: lay background for mathematical understanding of renormalization in the case of percolation, which might be easier, since no boundary conditions are involved.
Milestones: M VI.2
Addresses: Research Objective P1-P2.

Task VI.3. Percolation in physics.
·1 Partners 9 [UNIGE] and 4 [CNRS]: Elucidate some of physical questions (gradient percolation, self-avoiding walks, Brownian flights over polymers (typically DNA), to consider the widely open problem of DLA (diffusion limited aggregation) and some of its variants.
Milestones: M VI.2
Addresses: Research Objective P2-P3.

Task VI.4. Infinite dimensional systems.
·2 Partners 9 [UNIGE] and 3 [UH]: Investigate coupled lattice maps, and other more general infinite dimensional systems. Turbulent transport studied from the point of IFS approximations.
·3 Participation particular from the partner 4 [CNRS].
Milestones: M VI.2
Addresses: Research Objective P2-P3.