TOPOLOGICAL AND DYNAMICAL PROPERTIES IN PARAMETERIZED FAMILIES OF NON-HYPERBOLIC ATTRACTORS: THE INVERSE LIMIT APPROACH

National Science Center Poland, Sonata Bis grant 2020-2025

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TEAM

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DYI-SHING OU

Post-doc

Ph.D. Stony Brook University, N.Y. 
M.S. National Taiwan University, Taipei

MAGDALENA FORYŚ-KRAWIEC

Investigator

Ph.D. AGH University of Science and Tech.
M.Sc. Jagiellonian University

PRZEMEK KUCHARSKI

Ph.D. Student

M.Sc. Jagiellonian University

 
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The project studies parametric families of surface attractors, in parametric families of maps, such as Henon, Lozi, and Arnold families (and others). The focus is on the better understanding of bifurcations (transitions from simple to complex states) of these objects as parameter varies, from the point of view of their geometry (topology) and chaosity of dynamics. This includes continuity of the rotation number of accessible points in the attractors. The study employs the approach of inverse limits, that allow the study of complicated objects, such as fractal-like chaotic attractors, in terms of simpler objects, such as metric trees (dendrites), and their continuous transformations. The dynamical properties of the latter translate into dynamical properties of their natural extensions (shift homeomorphisms), which (semi)conjugate to strongly dissipative planar homeomorphisms. Another tool of interest is symbolic dynamics, which proved to be very useful in the case of multimodal interval maps (Milnor-Thurston kneading theory). However, a transition to dimension two has proved to be challenging so far, and therefore the development of appropriate analogue is of interest.

This project is funded by National Science Center (NCN), Poland in the years 2020-2025, under contract no. 2019/34/E/ST1/00237.

 

PROJECT PUBLICATIONS, PREPRINTS AND WORK IN PROGRESS

BORONSKI J., STIMAC SDENSELY BRANCHING TREES AS MODELS FOR HENON-LIKE AND LOZI-LIKE ATTRACTORS, SUBMITTED

BORONSKI J., CINC J., OPROCHA P.; BEYOND 0 AND ∞: A SOLUTION TO THE BARGE ENTROPY CONJECTURE, SUBMITTED

BORONSKI J., MINC, P., STIMAC S; ON CONJUGACY OF NATURAL EXTENSIONS OF 1-DIMENSIONAL MAPS, ERGODIC THEORY AND DYNAMICAL SYSTEMS, TO APPEAR

BORONSKI J.P., LINKED ORBITS OF HOMEOMORPHISMS OF THE PLANE AND GAMBAUDO-KOLEV THEOREM, SUBMITTED

OU D-SCRITICAL POINTS IN HIGHER DIMENSIONS, I: REVERSE ORDER OF PERIODIC ORBIT CREATIONS IN THE LOZI FAMILY, SUBMITTED

FORYŚ-KRAWIEC M., HANTAKOVA J., KUPKA J.,OPROCHA P., ROTH S., DENDRITES AND MEASURES WITH DISCRETE SPECTRUM, ERGODIC THEORY AND DYNAMICAL SYSTEMS DOI:10.1017/ETDS.2021.157

FORYŚ-KRAWIEC M., HANTAKOVA J., OPROCHA P.,ON THE STRUCTURE OF ALPHA-LIMIT SETS, DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS DOI:10.3934/DCDS.2021159

FORYŚ-KRAWIEC M., A FIXED POINT THEOREM FOR COMMUTING OPEN MAPS, SUBMITTED

KUCHARSKI P., ON CARTWRIGHT-LITTLEWOOD FIXED POINT THEOREM, INDAGATIONES MATHEMATICAE DOI:10.1016/J.INDAG.2022.02.009

KUCHARSKI P., GRAPH COVERS OF HIGHER DIMENSIONAL DYNAMICAL SYSTEMS, SUBMITTED

KUCHARSKI P.ATTRACTORS FOR THE FAMILY OF ORIENTATION PRESERVING LOZI MAPS, PREPRINT

BOROŃSKI J., OU D-S, DISCONTINUITIES IN PRIME END ROTATION NUMBERS OF HENON ATTRACTORS, IN PREPARATION

BORONSKI J., OU D-S, STIMAC S.,  KNEADING SEQUENCES FOR LOZI MAPS, IN PREPARATION

BORONSKI J., STIMAC S., THE CLASSIFICATION OF LOZI MAPS, IN PREPARATION 

BORONSKI J., CLARK A., KUCHERENKO T., ROTATION SETS OF PARAMETRIC FAMILIES OF ATTRACTORS ON THE TORUS, IN PROGRESS