We study the dynamics of the geodesic and horocycle flows of the unit tangent bundle $(hat M, T^1mathcal{F})$ 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.
We show that the horocycle flow associated with a foliation on a compact manifold by hyperbolic surfaces is minimal under certain conditions.
We show that the horocycle flows of open tight hyperbolic surfaces do not admit minimal sets.
We show that the equidistribution theorem of C. Bonatti and X. Gomez-Mont for a special kind of foliations by hyperbolic surfaces does not hold in general, and seek for a weaker form valid for general foliations by hyperbolic surfaces.
We shall show that the rotation of some irrational rotation number on the circle admits suspensions which are kinematic expansive.
Let $Pi_g$ be the surface group of genus $g$ ($ggeq2$), and denote by $RR_{Pi_g}$ the space of the homomorphisms from $Pi_g$ into the group of the orientation preserving homeomorphisms of $S^1$. Let $2g-2=kl$ for some positive integers $k$ and $l$. T
hen the subset of $RR_{Pi_g}$ formed by those $varphi$ which are semiconjugate to $k$-fold lifts of some homomorphisms and which have Euler number $eu(varphi)=l$ 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.
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 $r eq dim(M)+1$, then any homomorphism from $DC(M)_0$ to ${Diff}^1(R)$ or ${Diff}^1(S^1)$ is trivial.
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 sp
ectrum 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.
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 $S^m$ ($mgeq 2$), then any homomorphism from $DC(M)_0$ to $HR$ is trivial.
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 $ngeq2$, $r eq n+1$ and
$pgeq2$. 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 $rgeq3$.
