General Information:
This project is financed by The Executive Unit
for Financing Higher Education, Research, Development and Innovation (UEFISCDI), Romania, and has a duration of two years 2021-2022.
The host institution is the "Simion Stoilow" Institute of Mathematics of the Romanian Academy (IMAR).
Description:
Holomorphic dynamics in one or several variables is an exciting and
very active field of mathematics. In this project, we propose to study some new
classes of partially hyperbolic Hénon maps, in the real and in the
complex setting. We propose to identify some novel
regions of partially hyperbolicity, and thus expand the current
knowledge of the parameter space of complex Hénon maps.
We will
start by looking at the class of Hénon maps with a semi-parabolic fixed
(or periodic) point that come from small perturbations of quadratic
polynomials with a parabolic fixed (or periodic) point. By work of Radu
and Tanase we have a complete topological description for the
dynamics of these maps on their Julia sets and we know that these
maps lie in the boundary of hyperbolicity of the Hénon connectedness
locus. In this project we would like to show that they also belong to
the boundary of the complex horseshoe region. We know that we can
perturb these maps in nice ways in C2, so that we have continuity of
the Julia sets J and J+. We would now like to study the ergodic
properties of these maps as well, and investigate for example the
continuity of the Hausdorff dimension of their Julia sets with respect to
parameters. Furthermore, we would like to study partially hyperbolic
Hénon maps, which do not come from small perturbations. To this goal,
we propose to study a new class of Hénon maps with two semi-parabolic
basins, and understand the sequence
of tangencies that leads to the creation of the second semi-parabolic
basin.
This project is very ambitious, and requires new analytic
techniques, and a fine control of the weak expansion in the
semi-parabolic case. This project combines elements of Dynamical
Systems and Ergodic Theory, Analysis, Functional Analysis, Topology,
and Several Complex Variables.
Research Team:
- Raluca Tanase (Principal Investigator)
- Cristina Benea
- Remus Radu
Conference and Seminar Talks:
- Remus Radu, Dynamics and Computations Seminar, Uppsala University, Sweden, 10 May 2021
- Raluca Tanase, Dynamics and Computations Seminar, Uppsala University, Sweden, 18 May 2021
- Remus Radu, Colloquium Talk - IMAR Monthly Lecture, Bucharest, Romania, 19 May 2021
- Cristina Benea, Workshop for Young Researchers in Mathematics
(WYRM) 10th edition, 21 May 2021
- Raluca Tanase, Workshop for Young Researchers in Mathematics (WYRM) 10th edition, 21 May 2021
- Cristina Benea, Les journées du GdR AFHP 2021, Besançon, France, 27 - 29 September 2021
- Remus Radu, Conferinta Cercetarii Stiintifice din Academia Romana (CCSAR-2021),
Bucharest, Romania,
22 - 23 November 2021
- Raluca Tanase,
Connections Workshop: Complex Dynamics -
from special families to natural generalizations in one and several variables, MSRI, Berkeley, CA, USA, 2 - 4 February 2022
- Raluca Tanase, Arithmetic Dynamics International Online Seminar (ADIOS), organized by Cambridge University, United Kindom & York University, Canada, 12 April 2022
- Remus Radu, Workshop for Young Researchers in Mathematics (WYRM) 11th edition, 20 May 2022
- Raluca Tanase,
A XXIV-a Conferinta Nationala a Societatii de Stiinte Matematice din Romania,
University Politehnica Bucharest, Bucharest, Romania, 28 Octombrie 2022
- Raluca Tanase, Conference
Complex Dynamics in the Tropics - Celebrating Carsten Petersen's 60th birthday,
IMPA, Rio de Janeiro, Brazil, 27 November - 3 December 2022
- Raluca Tanase, Conference
Women in Mathematics in South-Eastern Europe,
ICMS-Sofia, Bulgarian Academy of Sciences, Sofia, Bulgaria, 8 - 9 December 2022
Other Conferences:
- Conference On Geometric Complexity of Julia sets III, Banach Center, Bedlewo, Poland,
26 September - 1 October 2021
-
Conference Advancing Bridges in Complex Dynamics, CIRM, Luminy, France, 20 - 24 September 2021
-
Conference Transcendental Dynamics and Beyond: Topics in Complex Dynamics, CRM, Barcelona,
Spain, 19 - 23 April 2021
-
Semester program Complex Dynamics -
from special families to natural generalizations in one and several variables, MSRI, Berkeley, CA, USA, 18 January - 27 May 2022
Papers and Preprints:
- T. Firsova, R. Radu, R. Tanase, Critical Locus for Henon maps in an HOV Region, arXiv:2211.12430
-
C. Benea, Y. Zhai, Multi-parameter flag Leibniz rules of arbitrary complexity in mixed-norm spaces, arXiv:2107.01426
-
R. Radu, R. Tanase, Complex Hénon maps with a semi-neutral fixed point, preprint 2021
-
R. Radu, R. Tanase, Global models for the Julia set J+, preprint 2021
-
R. Radu, R. Tanase, Semi-parabolic curves in the parameter space of complex Hénon maps, preprint 2022
-
R. Radu, R. Tanase, Horseshoes near semi-parabolic parameters, preprint 2022
Scientific Reports:
A scientific report can be found here (in Romanian).