On codimension two RR-closed foliations and group-actions

In this paper, we study $R$-closed foliations which are generalization of compact Hausdorff foliations. We show that the class space of a codimension-two-like $R$-closed foliation $\mathcal{F}$ (resp. group-action) on a compact connected manifold $M$ is a surface with conners, which is a generalization of the codimension two compact foliation cases, where the class space is a quotient space $M/\sim$ defined by $x \sim y$ if $\overline{\mathcal{F}(x)} = \overline{\mathcal{F}(y)}$ (resp. $\overline{O(x)} = \overline{O(y)}$). Moreover, a nontrivial $R$-closed flow on a connected compact $3$-manifold is either "almost two dimensional" or "almost one dimensional" or with "complicated" minimal sets. In addition, a homeomorphism on a hyperbolic compact surface $S$ is periodic if and only if it is $R$-closed (i.e. the orbit class space is Hausdorff). Each minimal set of an $R$-closed non-minimal non-periodic toral homeomorphism is a finite disjoint union of essential circloids

Recurrent and Non-wandering properties for foliations

In this paper, we define the recurrence and "non-wandering" for decompositions. The following inclusion relations hold for codimension one foliations on closed $3$-manifolds: $\{$minimal$\} \sqcup \{$compact$\}$ $\subsetneq$ $\{$pointwise almost periodic$\}$ $\subsetneq$ $\{$recurrent$\}$ $\subsetneq$ $\{$non-wandering$\}$ $\subsetneq$ $\{$Reebless$\}$. A non-wandering codimension one $C^2$ foliation on a closed connected $3$-manifold which has no leaf with uncountably many ends is minimal (resp. compact) if and only if it has no compact (resp. locally dense) leaves. In addition, the fundamental groups of all leaves of a codimension one transversely orientable $C^2$ foliation $\mathcal{F}$ on a closed $3$-manifold have the same polynomial growth if and only if $\mathcal{F}$ is without holonomy and has a leaf whose fundamental group has polynomial growth

Graph representations of surface flows

We construct a complete invariant for non-wandering surface flows with finitely many singular points but without locally dense orbits. Precisely, we show that a flow $v$ with finitely many singular points on a compact connected surface $S$ is a non-wandering flow without locally dense orbits if and only if $S/v_{\mathrm{ex}}$ is a non-trivial embedded multi-graph, where the extended orbit space $S/v_{\mathrm{ex}}$ is the quotient space defined by $x \sim y$ if they belong to either a same orbit or a same multi-saddle connection. Moreover, collapsing edges of the non-trivial embedded multi-graph $S/v_{\mathrm{ex}}$ into singletons, the quotient space $(S/v_{\mathrm{ex}})/\sim_E$ is an abstract multi-graph with the Alexandroff topology with respect to the specialization order. Therefore the non-wandering flow $v$ with finitely many singular points but without locally dense orbits can be reconstruct by finite combinatorial structures, which are the multi-saddle connection diagram and the abstract multi-graph $(S/v_{\mathrm{ex}})/\sim_E$ with labels. Moreover, though the set of topological equivalent classes of irrational rotations (i.e. minimal flows) on a torus is uncountable, the set of topological equivalent classes of non-wandering flows with finitely many singular points but without locally dense orbits on compact surfaces is enumerable by combinatorial structures algorithmically

R-closedness and Upper semicontinuity

Let $\mathcal{F}$ be a pointwise almost periodic decomposition of a compact metrizable space $X$. Then $\mathcal{F}$ is $R$-closed if and only if $\hat{\mathcal{F}}$ is usc. Moreover, if there is a finite index normal subgroup $H$ of an $R$-closed flow $G$ on a compact manifold such that the orbit closures of $H$ consist of codimension $k$ compact connected elements and "few singularities" for $k = 1$ or 2, then the orbit class space of $G$ is a compact $k$-dimensional manifold with conners. In addition, let $v$ be a nontrivial $R$-closed vector field on a connected compact 3-manifold $M$. Then one of the following holds: 1) The orbit class space $M/ \hat{v}$ is $[0,1]$ or $S^1$ and each interior point of $M/ \hat{v}$ is two dimensional. 2) $\mathrm{Per}(v)$ is open dense and $M = \mathrm{Sing}(v) \sqcup \mathrm{Per}(v)$. 3) There is a nontrivial non-toral minimal set. On the other hand, let $G$ be a flow on a compact metrizable space and $H$ a finite index normal subgroup. Then we show that $G$ is $R$-closed if and only if so is $H$

Preorder characterizations of lower separation axioms and their applications to foliations and flows

In this paper, we characterize several lower separation axioms $C_0, C_D$, $C_R$, $C_N$, $\lambda$-space, nested, $S_{YS}$, $S_{YY}$, $S_{YS}$, and $S_{\delta}$ using pre-order. To analyze topological properties of (resp. dynamical systems) foliations, we introduce notions of topology (resp. dynamical systems) for foliations. Then proper (resp. compact, minimal, recurrent) foliations are characterized by separation axioms. Conversely, lower separation axioms are interpreted into the condition for foliations and several relations of them are described. Moreover, we introduce some notions for topologies from dynamical systems and foliation theory

Genericity for non-wandering surface flows

Consider the set $\chi^0_{\mathrm{nw}}$ of non-wandering continuous flows on a closed surface. Then such a flow can be approximated by regular non-wandering flows without heteroclinic connections nor locally dense orbits in $\chi^0_{\mathrm{nw}}$. Using this approximation, we show that a non-wandering continuous flow on a closed connected surface is topologically stable if and only if the orbit space of it is homeomorphic to a closed interval. Moreover we state the non-existence of topologically stable non-wandering flows on closed surfaces which are not neither $\mathbb{S}^2$, $\mathbb{P}^2$, nor $\mathbb{K}^2$

Extended orbit properties on surfaces

In this paper, we study "demi-caract\'eristique" and (Poisson) stability in the sense of Poincar\'e. Using the definitions \'a la Poincar\'e for $\R$-actions $v$ on compact connected surfaces, we show that "$R$-closed" $\Rightarrow$ "pointwise almost periodicity (p.a.p.)" $\Rightarrow$ "recurrence" $\Rightarrow$ non-wandering. Moreover, we show that the action $v$ is "recurrence" with $|\mathrm{Sing}(v)| < \infty$ iff $v$ is regular non-wandering. If there are no locally dense orbits, then $v$ is "p.a.p." iff $v$ is "recurrence" without "orbits" containing infinitely singularities. If $|\mathrm{Sing}(v)| < \infty$, then $v$ is "$R$-closed" iff $v$ is "p.a.p."

Toral or non locally connected minimal sets for suspensions of RR-closed surface homeomorphisms

Let $M$ be an orientable connected closed surface and $f$ be an $R$-closed homeomorphism on $M$ which is isotopic to identity. Then the suspension of $f$ satisfies one of the following condition: 1) the closure of each element of it is minimal and toral. 2) there is a minimal set which is not locally connected. Moreover, we show that any positive iteration of an $R$-closed homeomorphism on a compact metrizable space is $R$-closed

Codimension one minimal foliations whose leaves have fundamental groups with the same polynomial growth

Let $\mathcal{F}$ be a transversely orientable codimension one minimal foliation without vanishing cycles of a manifold $M$. We show that if the fundamental group of each leaf of $\mathcal{F}$ has polynomial growth of degree $k$ for some non-negative integer $k$, then the foliation $\mathcal{F}$ is without holonomy

A topological characterization for non-wandering surface flows

Let $v$ be a continuous flow with arbitrary singularities on a compact surface. Then we show that if $v$ is non-wandering then $v$ is topologically equivalent to a $C^{\infty}$ flow such that there are no exceptional orbits and $\mathrm{P} \sqcup \mathop{\mathrm{Sing}}(v) = \{ x \in M \mid \omega(x) \cup \alpha(x) \subseteq \mathop{\mathrm{Sing}}(v) \}$, where $\mathrm{P}$ is the union of non-closed proper orbits and $\sqcup$ is the disjoint union symbol. Moreover, $v$ is non-wandering if and only if $\overline{\mathrm{LD}\sqcup \mathop{\mathrm{Per}}(v)} \supseteq M - \mathop{\mathrm{Sing}}(v)$, where $\mathrm{LD}$ is the union of locally dense orbits and $\overline{A}$ is the closure of a subset $A \subseteq M$. On the other hand, $v$ is topologically transitive if and only if $v$ is non-wandering such that $\mathop{\mathrm{int}}(\mathop{\mathrm{Per}}(v) \sqcup \mathop{\mathrm{Sing}}(v)) = \emptyset$ and $M - (\mathrm{P} \sqcup \mathop{\mathrm{Sing}}(v))$ is connected, where $\mathrm{int} {A}$ is the interior of a subset $A \subseteq M$. In addition, we construct a smooth flow on $\mathbb{T}^2$ with $\overline{\mathrm{P}} = \overline{\mathrm{LD}} =\mathbb{T}^2$