Braid invariant for periodic points of surface maps and its applications
Project/Area Number  09640115 
Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Geometry

Research Institution  Naruto University of Education 
Principal Investigator 
MATSUOKA Takashi Naruto University of Education, College of Education, Associate Professor, 学校教育学部, 助教授 (50127297)

CoInvestigator(Kenkyūbuntansha) 
HAYAKAWA Eijirou Toyama University, Faculty of Engineering, Associate Professor, 工学部, 助教授 (50240776)
MATSUNAGA Hiromichi Naruto University of Education, College of Education, Associate Professor, 学校教育学部, 教授 (30032634)

Project Period (FY) 
1997 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥2,400,000 (Direct Cost : ¥2,400,000)
Fiscal Year 1999 : ¥800,000 (Direct Cost : ¥800,000)
Fiscal Year 1998 : ¥700,000 (Direct Cost : ¥700,000)
Fiscal Year 1997 : ¥900,000 (Direct Cost : ¥900,000)

Keywords  periodic point / 2dimensional embedding / braid / fixed point index / pseudoAnosov map / unstable fixed point / 不動点 / 埋め込み写像 / 組ひも / 2次元力学系 / 位相的エントロピー / 絡み数 
Research Abstract 
We studied the topological structure of the set of periodic points for embeddings of a 2dimensional disk to itself, by exploiting the braid invariant, which is one of the topological invariants defined for periodic points. Our results are the following : 1. Thurston's theory has shown that if the embedding has a periodic point which is topologically complicated (i.e., its braid invariant has a pseudoAnosov component in its decomposition), then there exist infinitely many periodic points. We proved that if P(n), the set of periodic points with period less than or equal to an integer n, is topologically complicated, then P(n) has at least 2n+3 points. 2. When P(n) has at most 2n+2 points, the above result shows that this set is topologically simple. In this case, we determined all the possible forms of the braid invariant of P(n). 3. We studied fixed points from a viewpoint different from the above, and obtained the following : (1) Using the notion of a braid, we introduced an equivalence relation on the fixed point set, and proved that the braid invariant of each equivalence class is of a simple type. This implies that the study of the structure of the fixed point set is divided into two parts: the study of the property of each equivalence class and that of how the equivalence classes are combined together. (2) We proved that the fixed point index of each equivalence class is less than two. As an application of this result, we studied the relationship between the topological property and stability of fixed points, and showed that every equivalence class with at least two points must contain an unstable fixed point. Moreover, we showed that the number of equivalence classes having an unstable fixed point is greater than that of the equivalence classes containing no unstable fixed points.

Report
(4results)
Research Output
(13results)