Shigenori Matsumoto

• Nihon University

Denote by $\DC(M)_0$ the identity component of the group of the compactly supported $C^r$ diffeomorphisms of a connected $C^\infty$ manifold $M$. We show that if $\dim(M)\geq2$ and $r\neq \dim(M)+1$, then any homomorphism from $\DC(M)_0$ to ${\Diff}^1(\R)$ or ${\Diff}^1(S^1)$ is trivial.

We give a shorter proof of the following theorem of K. Mann [Homomorphisms between diffeomorphism groups; arxiv 1206:1196]: the identity component of the group of the compactly supported C r diffeomorphisms of ℝ n cannot admit a nontrivial C p -action on S 1 , provided n≥2,r≠n+1 and p≥2. We also give a new proof of another theorem of Mann [loc. cit...

Let $\FF$ be a codimension one foliation on a closed manifold $M$ which admits a transverse dimension one Riemannian foliation. Then any continuous leafwise harmonic functions are shown to be constant on leaves.

We consider an orientation preserving homeomorphism $h$ of $S^2$ which admits a repellor denoted $\infty$ and an attractor $-\infty$, which is not a North-South map, such that the basins of $\infty$ and $-\infty$ intersect. We study various aspects of the rotation number of $h:S^2\setminus\{\pm\infty\}\to S^2\setminus\{\pm\infty\}$, especially its...

The first half of this paper is concerned with the topology of the space $\AAA(M)$ of (not necessarily contact) Anosov vector fields on the unit tangent bundle $M$ of closed oriented hyperbolic surfaces $\Sigma$. We show that there are countably infinite connected components of $\AAA(M)$, each of which is not simply connected. In the second part, w...

This paper is concerned about the orbit equivalence types of $C^\infty$ diffeomorphisms of $S^1$ seen as nonsingular automorphisms of $(S^1,m)$, where $m$ is the Lebesgue measure. Given any Liouville number $\alpha$, it is shown that each of the subspace formed by type ${\rm II}_1$, ${\rm II}_\infty$, ${\rm III}_\lambda$ ($\lambda>1$), ${\rm III}_\...

Given any Liouville number $\alpha$, it is shown that various subspaces are $C^\infty$-dense in the space of the orientation preserving $C^\infty$ diffeomorphisms of the circle with rotation number $\alpha$.

We consider a fixed point free homeomorphsim $h$ of the closed band $B=\R\times[0,1]$ which leaves each leaf of a Reeb foliation on $B$ invariant. Assuming $h$ is the time one of various topological flows, we compare the restriction of the flows on the boundary.

We show that there are no normally contracting actions of unimodular Lie groups on closed manifolds.

Among the topological conjugacy classes of the continuous flows $\{\phi^t\}$ whose orbit foliations are the planar Reeb foliation, there is one class called the standard Reeb flow. We show that $\{\phi^t\}$ is conjugate to the standard Reeb flow if and only if $\{\phi^t\}$ is conjugate to $\{\phi^{\lambda t}\}$ for any $\lambda>0$.

We consider the rotation number $\rho(t)$ of a diffeomorphism $f_t=R_t\circ f$, where $R_t$ is the rotation by $t$ and $f$ is an orientation preserving $C^\infty$ diffeomorphism of the circle $S^1$. We shall show that if $\rho(t)$ is irrational $$\limsup_{t'\to t}(\rho(t')-\rho(t))/(t'-t)\geq 1.$$

Given any Liouville number $\alpha$, it is shown that the nullity of the Hausdorff dimension of the invariant measure is generic in the space of the orientation preserving $C^\infty$ diffeomorphisms of the circle with rotation number $\alpha$.

For any irrational number $\alpha$, there exists an ergodic area preserving homeomorphism of the closed annulus which is isotopic to the identitity, admits no compact invariant set contained in the interior of the annulus, and has the rotation number $\alpha$. Comment: 10 pages

Let $f$ be a homeomorphism of the closed annulus $A$ isotopic to the identity, and let $X\subset {\rm Int}A$ be an $f$-invariant continuum which separates $A$ into two domains, the upper domain $U_+$ and the lower domain $U_-$. Fixing a lift of $f$ to the universal cover of $A$, one defines the rotation set $\tilde \rho(X)$ of $X$ by means of the i...

We construct, for each irrational number $\alpha$, a minimal $C^1$-diffeomorphism of the circle with rotation number $\alpha$ which admits a measur

Let $f$ be an orientation preserving homeomorphism of $S^2$ which has a (nontrivial) continuum $X$ as a minimal set. Then there are exactly two connected components of $S^2\setminus X$ which are left invariant by $f$ and all the others are wandering. The Carath\'eodory rotation number of an invariant component is irrational.

We show that there are no Anosov actions by (n-1)-dimensional unimodular Lie groups on closed n-dimensional manifolds.

A flow defined by a nonsingular smooth vector field $X$ on a closed manifold $M$ is said to be parameter rigid if given any real valued smooth function $f$ on $M$, there are a smooth funcion $g$ and a constant $c$ such that $f=X(g)+c$ holds. We show that the parameter rigid flows on closed orientable 3-manifolds are smoothly conjugate to Kronecker...

We prove that a transversely equicontinuous minimal lamination on a locally compact metric space $Z$ has a transversely invariant Radon measure. Moreover if the space $Z$ is compact, then the tranversely invariant Radon measure is shown to be unique up to a scaling.

Relations between parameter rigidity of locally free Lie group actions on closed manifolds and the 1st leafwise cohomology of the orbit foliations are discussed. Some computational results of the leafwise cohomology are included. Comment: 21 pages

We show that a so called split Anosov $R^{n}$ -action on a closed oriented $(2n+1)$ -dimensional manifold is $C^{\infty}$ conjugate to the suspension of a split hyperbolic alfine representation of $Z^{n}$ on the $(n+1)$ -dimensional torus.

Given an isolated fixed point p of index one, of an area- and orientation-preserving homeomorphism f of a surface, we define a type (positive, negative) for a lift of f to the universal cover of D\setminus \{p\}, where D is a small disc neighbourhood of p. Using this concept, we establish a new kind of fixed point theorem. Furthermore, a refinement...

We consider the rotation setR of a homeomorphismf, isotopic to the identity, of a closed surface of genusg2. We show if Int(R) is nonempty and contains an element which is realized by an asymptotic measure, then all the rational points of Int(R) are realized by periodic orbits. We raise an example to show that the second condition above is indispen...

The goal of this paper is to give, under some hypotheses, a characterization of currents and distributions invariant by a group of diffeomorphisms of a manifold M and especially in the case of a Kleinian group Γ acting on the n-sphere Sⁿ.

Without Abstract

Consider a nonsingular vector field X on a closed manifold M n . As a matter of fact, X always admits a transverse codimension one plane field, which however may fail to be integrable. In fact it is well known that there are many examples of vector fields which do not admit transverse foliations.

Every two homomorphisms from the fundamental group of an oriented closed surface of genus ≧2 into the group of orientation preserving homeomorphisms (resp.C 2 diffeomorphisms) of the circle are shown to be mutually semi-conjugate (resp. topologically conjugate), provided their Euler numbers attain the minimal value (or the maximal value) allowed by...

We consider the condition when bounded cohomology injects into ordinary cohomology and prove the vanishing of bounded cohomology of the group of all compactly supported homeomorphisms of Rn.

Without Abstract

1. Let M" be an n-dimensional closed C~-manifold. The space ~r of C r diffeomorphisms of M" with the C r topology is a C ~ Banach manifold for 1 2. Given two isotopic C ~ Morse-Smale (M-S) diffeomorphisms of M", does there exist an arc ~ ~ ~o,, joining them which has only finitely many (or countably many) bifurcations?