Anosov actions of (n-1)-dimensional Lie groups on n-dimensional manifolds
We show that there are no Anosov actions by (n-1)-dimensional unimodular Lie groups on closed n-dimensional manifolds.
math.DS↗
arXiv subjects
Shigenori Matsumoto
Publications and source records attributed to Shigenori Matsumoto.
Normally contracting Lie group actions
We show that there are no normally contracting actions of unimodular Lie groups on closed manifolds.
math.DS↗
Two remarks on $C^\infty$ Anosov diffeomorphisms
Let $M$ be a closed oriented $C^\infty$ manifold and $f$ a $C^\infty$ Anosov diffeomorphism on $M$. We show that if $M$ is the two torus $T^2$, then $f$ is conjugate to a hyperbolic automorphism of $T^2$, either by a $C^\infty$ diffeomorphism or by a singular homeomorphism. We also show that for general $M$, if $f$ admits an absolutely continuous invariant measure $μ$, then $μ$ is a $C^\infty$ volume. The proofs are concatenations of well known results in the field.
math.DS↗
A generic dimensional property of the invariant measures for circle diffeomorphisms
Given any Liouville number $α$, 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 $α$.
math.DS↗
The dichotomy of harmonic measures of compact hyperbolic laminations
Given a harmonic measure of a hyperbolic lamination on a compact metric space, a positive harmonic function is defined on the universal cover of a typical leaves. We discuss some properties of this function. Especially if all the leaves are hyperbolic, ergodic harmonic measures are divided into two classes.
math.GT↗
Dense properties of the space of the circle diffeomorphisms with a Liouville rotation number
Given any Liouville number $α$, 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 $α$.
math.DS↗
A characterization of the standard Reeb flow
Among the topological conjugacy classes of the continuous flows $\{ϕ^t\}$ whose orbit foliations are the planar Reeb foliation, there is one class called the standard Reeb flow. We show that $\{ϕ^t\}$ is conjugate to the standard Reeb flow if and only if $\{ϕ^t\}$ is conjugate to $\{ϕ^{λt}\}$ for any $λ>0$.
math.DS↗
Rotation sets of invariant separating continua of annular homeomorphisms
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 $\tilde ρ(X)$ of $X$ by means of the invariant probabilities on $X$. For any rational number $p/q\in \tildeρ(X)$, $f$ is shown to admit a $p/q$ periodic point in $X$, provided that (1) $X$ consists of nonwandering points or (2) $X$ is an attractor and the frontiers of $U_-$ and $U_+$ coincides with $X$. Also the Carathéodory rotation numbers of $U_\pm$ are shown to be in $\tildeρ(X)$ for any separating invariant continuum $X$.
math.DS↗
Continua as minimal sets of homeomorphisms of S^2
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 Carathéodory rotation number of an invariant component is irrational.
math.DS↗
Minimal C^1 diffeomorphisms of the circle which admit measurable fundamental domains
We construct, for each irrational number $α$, a minimal $C^1$-diffeomorphism of the circle with rotation number $α$ which admits a measur
math.DS↗
Flows of flowable Reeb homeomorphisms
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.
math.DS↗
Derivatives of rotation number of one parameter families of circle diffeomorphisms
We consider the rotation number $ρ(t)$ 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 $ρ(t)$ is irrational $$\limsup_{t'\to t}(ρ(t')-ρ(t))/(t'-t)\geq 1.$$
math.DS↗
Annular area preserving homeomorphisms which admit no interior compact invariant sets
For any irrational number $α$, 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 $α$.
math.DS↗
Heat Transport in a Random Packing of Hard Spheres
Heat conduction in a random packing of hard spheres is studied by nonequilibrium molecular dynamics simulation. We find a hard-sphere random packing shows higher thermal conductivity than a crystalline packing with same packing fraction. Under the same pressure, the random structure causes reduction of thermal conductivity by only 10% from crystalline packing, which is consistent with the experimental fact that amorphous materials can have high thermal conductivity which is comparable to that of crystals.
cond-mat.stat-mech↗
Prime end rotation numbers of invariant separating contunua of annular homeomorphisms
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 $\tilde ρ(X)$ of $X$ by means of the invariant probabilities on $X$, as well as the prime end rotation number $\checkρ_\pm$ of $U_\pm$. The purpose of this paper is to show that $\checkρ_\pm$ belongs to $\tildeρ(X)$ for any separating invariant continuum $X$.
math.DS↗
Rigidity of locally free Lie group actions and leafwise cohomology
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.
math.GT↗
The parameter rigid flows on oriented 3-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 flows on the 3-torus with badly approximable slope.
math.GT↗
The unique ergodicity of equicontinuous laminations
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.
math.GT↗