Topological dynamical system on surfaces

• Grant number: 18/03762-2
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: June 01, 2018
• End date: March 08, 2022
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Fábio Armando Tal
• Grantee: Xiaochuan Liu
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Associated research grant: 16/25053-8 - Dynamics and geometry in low dimensions, AP.TEM

Abstract

The main focus is to study rotation theory on closed surfaces.Rotation theory aims to understand dynamical system on surfaces. The field provides with lots of questions and stimulates many deep progresses in recent years in the theory of surface topological dynamics. In the case of two-torus, we define a rotation set. One important issue is to study the shape of a rotation set. Another direction is the study of how rotation sets for torus homeomorphisms vary in parameterised family of homeomorphisms. For example, we will study area-preserving families. We plan to show or disprove that, for the families of conservative homeomorphisms, the set of parameter for which the rotation set has empty interior is connected.We try to apply techniques of the forcing paper of P. Le calvez and F.Tal to derive estimates for the lower bound of the entropy of torus homeomorphisms, which depends only on the rotation set of the maps. This was already started by Kwapisz in his thesis but the result should be improved using modern methods. In a parameterised family, we want to give a good description of the dynamics for parameters where the rotation set is not constant. At the very least we want to understand the boundary of the connected components where the rotation remains constant.More detailes will explained in the document "Plano de atividades da bolsa" attached in this site.

Scientific publications

(7)

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

KLEIN, SILVIUS; LIU, XIAO-CHUAN; MELO, ALINE. Uniform convergence rate for Birkhoff means of certain uniquely ergodic toral maps. Ergodic Theory and Dynamical Systems, v. 41, n. 11, p. 3363-3388, NOV 2021. (18/03762-2)

BORONSKI, JAN P.; KENNEDY, JUDY; LIU, XIAO-CHUAN; OPROCHA, PIOTR. Minimal Non-invertible Maps on the Pseudo-Circle. Journal of Dynamics and Differential Equations, v. 33, n. 4, p. 1897-1916, DEC 2021. (18/03762-2)

ADDAS-ZANATA, SALVADOR; LIU, XIAO-CHUAN. On stable and unstable behaviour of certain rotation segments. Nonlinearity, v. 35, n. 11, p. 39-pg., 2022-11-03. (18/03762-2)

LIU, XIAO-CHUAN; YANG, XU. ON THE TUR \'AN NUMBER OF GENERALIZED THETA GRAPHS. SIAM JOURNAL ON DISCRETE MATHEMATICS, v. 37, n. 2, p. 15-pg., 2023-01-01. (18/03762-2)

BORONSKI, JAN P.; KENNEDY, JUDY; LIU, XIAO-CHUAN; OPROCHA, PIOTR. Minimal Non-invertible Maps on the Pseudo-Circle. Journal of Dynamics and Differential Equations, AUG 2020. (18/03762-2)

BORONSKI, JAN P.; CINC, JERNEJ; LIU, XIAO-CHUAN. Prime ends dynamics in parametrised families of rotational attractors. JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES, APR 2020. (18/03762-2)

BORONSKI, JAN P.; CINC, JERNEJ; LIU, XIAO-CHUAN. Prime ends dynamics in parametrised families of rotational attractors. JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES, v. 102, n. 2, p. 23-pg., 2020-04-07. (18/03762-2)