Topological invariants for periodic points of torus maps
Project/Area Number 
13640079

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, Professor, 学校教育学部, 教授 (50127297)

CoInvestigator(Kenkyūbuntansha) 
HAYAKAWA Eijirou Toyama University, Faaculty of Engineering, Associate Professor, 工学部, 助教授 (50240776)
MURATA Hiroshi Naruto University of Education, College of Education, Professor, 学校教育学部, 教授 (20033897)
NARUKAWA Kimiaki Naruto University of Education, College of Education, Professor, 学校教育学部, 教授 (60116639)

Project Period (FY) 
2001 – 2002

Project Status 
Completed (Fiscal Year 2002)

Budget Amount *help 
¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2002: ¥700,000 (Direct Cost: ¥700,000)

Keywords  Fixed point index / torus map / braid / surface homeomorphism / unstable fixed point / topological entropy / 曲面 / 組ひも / トーラス 
Research Abstract 
We considered a homeomorphism on a compact orientable surface which is isotopic to the identity map. Using fixed point index and braid invariant, we investigated the topological property of periodic points of this map, and has obtained the following results on fixed points: 1. We introduced a relation on the fixed point set, and showed that this relation is an equivalence relation. 2. We proved that the fixed point index of an equivalence class is invariant under the isotopy of the map. This is useful to determine the fixed point indices of equivalence classes of given map 3. We showed that every equivalence class has fixed point index at most one 4. Applying the above result, we showed that if an equivalence class has at least two fixed points, then one of them must be unstable. This result gives a relationship between the stability of a fixed point and the global topology of the map 5. The number of equivalence classes containing an unstable point is greater than half of the number of all equivalence classes minus the genus of the surface 6. If these is an equivalence class which has only one point and this point has index one, then the topological entropy of the map is positive. This shows that a topological property of fixed points implies the dynamical property of the map

Report
(3 results)
Research Products
(4 results)