2005 Fiscal Year Final Research Report Summary
Exceptional minimal sets of codimension two on flows
| Project/Area Number |
15540078
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | HIROSHIMA UNIVERSITY |
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Principal Investigator |
NAKAYAMA Hiromichi Hiroshima University, Faculty of Integrated Arts and Sciences, Associate Professor, 総合科学部, 助教授 (30227970)
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| Co-Investigator(Kenkyū-buntansha) |
MATSUMOTO Shigenori Nihon University, College of Science and Technology, Professor, 理工学部, 教授 (80060143)
INABA Takashi Chiba University, Graduate School of Science and Technology, Professor, 大学院・自然科学研究科, 教授 (40125901)
SHIBATA Tetsutaro Hiroshima University, Graduate Scholl of Engineering, Professor, 大学院・工学研究科, 教授 (90216010)
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| Project Period (FY) |
2003 – 2005
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| Keywords | topological dynamics / exceptional minimal set |
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| Research Abstract |
1.Construction of surface diffeomorphisms with exceptional minimal sets
In 1995, McSwiggen constructed a surface diffeomorphism with a locally connected exceptional minimal set. He inserted a cylindrical hole along a leaf of the strong unstable foliation of a 3-dimensional hyperbolic toral automorphism. We constructed a new example from a 3-dimensional projectively Anosov diffeomorphism.
2.Exceptional minimal sets for group actions on surfaces
In particular, we studied exceptional minimal sets of group actions on the sphere. In the case of hyperbolic groups, it is well known that there are many shapes of exceptional minimal sets. Then we consider the problem what is the easiest group which acts on a surface with a complicated minimal sets. We showed that the exceptional minimal set of an action of a nilpotent group is not homeomorphic to the Siepinski carpet. Furthermore, in the case of the fundamental groups of torus bundles over the circle, we obtain some partial results.
3.Minimal flows of 3-dimensional manifolds
On minimal flows, we investigated their infinitesimal flows and projective flows, we obtain the following dynamical properties :
(1)We used Zimmer's theory in order to obtain invariant fiber measures of projective flows, and deduced several topologaical properties of the original flows.
(2)We investigated minimal flows whose projective flows have exactly two minimal sets, and we showed that it is topologically equivalent to an irrational flow if its projective flow has more than three minimal sets. Furthermore, there is a bundle section separating the minimal sets if there are exactly two minimal sets.
(3)Contreras obtained important results on weakly partially hyperbolic flows. We used his theory to the projective flows and proved the existence of the "upper" orbits and the "lower" orbits of the projective flow if it has exactly two minimal sets.
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Research Products
(50 results)
