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Shigenori Matsumoto

Publications and source records attributed to Shigenori Matsumoto.

At least 19 recordsLinked to original sources
arXiv2018

Dynamics of isolated orders

We study some properties of the dynamical realization of isolated left orders of a countable group G. We show that the dynamical realization admits a unique minimal set provided G is not infinite cyclic. We show that convex subgroup of an isolated left order is finite in number. Using this, we give a dynamical proof of the Tararin theorem. We also show that there is a new isolated order on the braid group B_3.

math.GR↗
arXiv2017

Isolated circular orders of PSL(2,Z)

We give a bijection between the isolated circular orders of the group G=PSL(2,Z) and the equivalence classes of Markov systems associated with G. As applications, we present examples of isolated circular order of G.

math.DS↗
arXiv2015

Basic partitions and combinations of group actions on the circle: A new approach to a theorem of Kathryn Mann

Let $Π_g$ be the surface group of genus $g$ ($g\geq2$), and denote by $\RR_{Π_g}$ the space of the homomorphisms from $Π_g$ into the group of the orientation preserving homeomorphisms of $S^1$. Let $2g-2=kl$ for some positive integers $k$ and $l$. Then the subset of $\RR_{Π_g}$ formed by those $φ$ which are semiconjugate to $k$-fold lifts of some homomorphisms and which have Euler number $eu(φ)=l$ is shown to be clopen. This leads to a new proof of the main result of Kathryn Mann \cite{Mann} from a completely different approach.

math.DS↗
arXiv2015

Horocycle flows for laminations by hyperbolic Riemann surfaces and Hedlund's theorem

We study the dynamics of the geodesic and horocycle flows of the unit tangent bundle $(\hat M, T^1\mathcal{F})$ of a compact minimal lamination $(M,\mathcal F)$ by negatively curved surfaces. We give conditions under which the action of the affine group generated by the joint action of these flows is minimal, and examples where this action is not minimal. In the first case, we prove that if $\mathcal F$ has a leaf which is not simply connected, the horocyle flow is topologically transitive.

math.DS↗
arXiv2014

A glimpse of fluid turbulence from the molecular scale

Large scale molecular dynamics simulations of freely decaying turbulence in three-dimensional space are reported. Fluid components are defined from the microscopic states by eliminating thermal components from the coarse-grained fields. The energy spectrum of the fluid components is observed to scale reasonably well according to Kolmogorov scaling determined from the energy dissipation rate and the viscosity of the fluid, even though the Kolmogorov length is of the order of the molecular scale.

cond-mat.stat-mech↗
arXiv2013

Actions of groups of diffeomorphisms on one-manifolds

Denote by $\DC(M)_0$ the identity component of the group of compactly supported $C^\infty$ diffeomorphisms of a connected $C^\infty$ manifold $M$, and by $\HR$ the group of the homeomorphisms of $\R$. We show that if $M$ is a closed manifold which fibers over $S^m$ ($m\geq 2$), then any homomorphism from $\DC(M)_0$ to $\HR$ is trivial.

math.GT↗
arXiv2013

New proofs of theorems of Kathryn Mann

We give a shorter proof of the following theorem of Kathryn Mann \cite{M}: 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 $n\geq2$, $r\neq n+1$ and $p\geq2$. We also give a new proof of another theorem of Mann: any nontrivial endomorphism of the group of the orientation preserving $C^r$ diffeomorphisms of the circle is the conjugation by a $C^r$ diffeomorphism, if $r\geq3$.

math.GT↗
arXiv2013

Nontrivial attractor-repellor maps of $S^2$ and rotation numbers

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 relationship with the existence of periodic orbits.

math.DS↗
arXiv2012

The space of (contact) Anosov flows on 3-manifolds

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 $Σ$. We show that there are countably infinite connected components of $\AAA(M)$, each of which is not simply connected. In the second part, we study contact Anosov flows. We show in particular that the time changes of contact Anosov flows form a $C^1$-open subset of the space of the Anosov flows which leave a particular $C^\infty$ volume form invariant, if the ambiant manifold is a rational homology sphere.

math.DS↗
arXiv2012

Orbit equivalence types of circle diffeomorphisms with a Liouville rotation number

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 $α$, it is shown that each of the subspace formed by type ${\rm II}_1$, ${\rm II}_\infty$, ${\rm III}_λ$ ($λ>1$), ${\rm III}_\infty$ and ${\rm III}_0$ diffeomorphisms are $C^\infty$-dense in the space of the orientation preserving $C^\infty$ diffeomorphisms with rotation number $α$.

math.DS↗