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Tomoo Yokoyama

Publications and source records attributed to Tomoo Yokoyama.

At least 37 records · Page 2Linked to original sources
arXiv2022

Morse hyper-graphs and abstract weak orbit spaces of semi-decompositions

We introduce topological invariants of semi-decompositions (e.g. filtrations, semi-group actions, multi-valued dynamical systems, combinatorial dynamical systems) on a topological space to analyze semi-decompositions from a dynamical systems point of view. In fact, we construct Morse hyper-graphs and abstract weak elements of semi-decompositions. Moreover, the Morse hyper-graphs of the set of sublevel sets of a Morse function of a compact manifold is the Reeb graph of such function as abstract multi-graphs. The abstract weak element space for a simplicial complex is the face poset. The abstract weak element spaces of positive orbits of acyclic directed graphs are their abstract directed multi-graphs.

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arXiv2022

Topological characterizations of Hamiltonian flows with finitely many singular points on unbounded surfaces

Hamiltonian flows on compact surfaces are characterized, and the topological invariants of such flows with finitely many singular points are constructed from the viewpoints of integrable systems, fluid mechanics, and dynamical systems. Though various fluid phenomena are modeled as flows on the plane, it is not obvious to determine if the flows are Hamiltonian, even the singular point set is totally disconnected and every orbit is contained in a straight line parallel to the $x$-axis. In fact, there are such non-Hamiltonian flows on the plane. On the other hand, this paper topologically characterizes Hamiltonian flows on unbounded surfaces and constructs their complete invariant under a regularity condition for singular points. In addition, under finite volume assumption, Hamiltonian flows on unbounded surfaces can be embedded into those on compact surfaces.

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arXiv2022

Morse hyper-graphs of topological spaces and decompositions

The cell complex structure is one of the most fundamental structures in topology and combinatorics, the Morse decomposition of a dynamical system analyzes the global gradient behavior, and the Reeb graph of a function is an elementary tool in Morse theory to represent the global connection and is also used to analyze continuous and discrete data in topological data analysis. In this paper, we unify three concepts of cell complexes, Morse decompositions, and Reeb graphs into a concept. In fact, we introduce topological invariants for topological spaces and decompositions, which are analogous to abstract (weak) orbit spaces and Morse graphs for flows. To achieve these, we define analogous concepts of recurrence and "chain-recurrence" for topological spaces and decompositions such that the original and new recurrences correspond to each other for a flow orbits on a locally compact Hausdorff space and the orbit space. We show that the Morse hyper-graphs exist for any topological spaces and any invariant decompositions (e.g. compact foliated spaces). Moreover, the abstract weak element spaces generalize the Reeb graphs of Morse functions and the Morse (hyper-)graphs and refine abstract cell complexes.

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arXiv2022

Flows with time-reversal symmetric limit sets on surfaces

The Long-time behavior of orbits is one of the most fundamental properties in dynamical systems. Poincaré studied the Poisson stability, which satisfies a time-reversal symmetric condition, to capture the property of whether points return arbitrarily near the initial positions after a sufficiently long time. Birkhoff introduced and studied the concept of non-wandering points, which is one of the time-reversal symmetric conditions. Moreover, minimality and pointwise periodicity satisfy the time-reversal symmetric condition for limit sets. This paper characterizes flows with the time-reversal symmetric condition for limit sets, which refine the characterization of irrational or Denjoy flows by Athanassopoulos. Using the description, we construct flows on a sphere with Lakes of Wada attractors and with an arbitrarily large number of complementary domains, which are flow versions of such examples of spherical homeomorphisms constructed by Boroński, Činč, and Liu.

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arXiv2022

On low-codimensional decompositions

Decompositions on manifolds appear in various geometric structures. Necessary and sufficient conditions for quotient spaces of decompositions to be manifolds are widely characterized. We characterize necessary and sufficient conditions to be $k$-manifolds $(k = 1, 2)$, which generalize characterizations in the codimension-$k$ cases for the leaf spaces of foliations, the orbit spaces of group-actions, decomposition spaces of upper semi-continuous decompositions, and leaf class spaces of Riemannian foliation with regular closure. To prove such characterizations, we generalize a characterization of upper semi-continuity for decomposition into one for a class decomposition. In addition, we completely characterize upper semi-continuity for class decompositions of homeomorphisms on orientable compact surfaces and of homeomorphisms isotopic to identity on non-orientable compact surfaces. Furthermore, a flow on a connected compact $3$-manifold whose class decomposition is upper semi-continuous is "almost $k$ dimensional" $(k=0, 1, 2, 3)$ or has "complicated" minimal sets.

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arXiv2022

Generalization of Poincaré recurrence theorem for flows on surfaces and characterization of minimal flows on compact surfaces

Poincaré recurrence theorem implies the density of recurrent points for volume-preserving dynamical systems on compact domains. The density of closed orbits in the non-wandering set is one of the essential properties of Axiom A and chaos. The minimal flows are one of the most fundamental objects in topological dynamics. To analyze minimality, recurrence, and density in this paper, we introduce a strict limit circuit and a circuit with wandering holonomy. Using these concepts, we classify non-recurrent non-wandering orbits of flows on compact surfaces with finitely many singular points, characterize the minimality of flows on compact surfaces, and generalize the Poincaré recurrence theorem for flows on surfaces. Moreover, we characterize the density of closed orbits in the non-wandering set under finiteness of singular points. Furthermore, we demonstrate the necessity of finiteness of singular points and compactness of surfaces. In addition, the analogous results hold for flows with finitely many connected components of the singular point set on non-compact surfaces using the end compactification and collapsing singular points.

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arXiv2021

Topological characterizations of Morse-Smale flows on surfaces and generic non-Morse-Smale flows

It is known that $C^r$ Morse-Smale vector fields form an open dense subset in the space of vector fields on orientable closed surfaces and are structurally stable for any $r \in \mathbb{Z}_{>0}$. In particular, $C^r$ Morse vector fields (i.e. Morse-Smale vector fields without limit cycles) form an open dense subset in the space of $C^r$ gradient vector fields on orientable closed surfaces and are structurally stable. Therefore generic time evaluations of gradient flows on orientable closed surfaces (e.g. solutions of differential equations) are described by alternating sequences of Morse flows and instantaneous non-Morse gradient flows. To illustrate the generic transitions, we characterize and list all generic non-Morse gradient flows. To construct such characterizations, we characterize isolated singular points of gradient flows on surfaces. In fact, such a singular point is a non-trivial finitely sectored singular point without elliptic sectors. Moreover, considering Morse-Smale flows as "generic gradient flows with limit cycles", we characterize and list all generic non-Morse-Smale flows.

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arXiv2021

Topological Bifurcation Structure Of One-Parameter Families Of $C^1$ Unimodal Maps

We consider the bifurcation structure of one-parameter families of unimodal maps whose differentiability is only $C^1$. The structure of its bifurcation diagram can be a very wild one in such case. However we prove that in a certain topological sense, the structure is the same as that of the standard family of quadratic polynomials. In the case of families of polynomials, irreducible component of the bifurcation diagram can be defined naturally by dividing by the polynomials corresponding to lower periods. We show that such a irreducible component can be defined even if the maps and the family satisfy only a very mild differentiability condition. By removing components of lower periods, the structure of the bifurcation diagram becomes a considerably simplified one. We prove that the symbolic condition for the irreducible components is exactly the same as that of the standard family of quadratic polynomials. When we consider families of maps without strict conditions, the bifurcation diagrams may have infinitely many wild components. We show that such a situation does not affect the irreducible component essentially by proving a separation theorem for compact set in the plane which asserts that a given connected component can be cut out from the compact set by a curve.

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arXiv2020

Quotient spaces and topological invariants of flows

We construct topological invariants, called abstract weak orbit spaces, of flows and homeomorphisms on topological spaces, to describe both gradient dynamics and recurrent dynamics. In particular, the abstract weak orbit spaces of flows on topological spaces are generalizations of both Morse graphs of flows on compact metric spaces and Reeb graphs of Hamiltonian flows with finitely many singular points on surfaces. Moreover, we show that the abstract weak orbit spaces are complete and finite for several kinds of flows on manifolds, and we state several examples whose Morse graphs are singletons but whose abstract weak orbit spaces are not singletons. In addition, we consider when the time-one map reconstructs the topology of the original flow. Therefore we show that the orbit space of a Hamiltonian flow with finitely many singular points on a compact surface is homeomorphic to the abstract weak orbit space of the time-one map by taking an arbitrarily small reparametrization, and that the abstract weak orbit spaces of a Morse flow on a compact manifold and the time-one map are homeomorphic. Furthermore, the abstract weak orbit space of a Morse flow on a closed manifold is a refinement of the CW decomposition which consists of the unstable manifolds of singular points. Though the CW decomposition of a Morse flow on a closed manifold is finite, the intersection of the unstable manifold and the stable manifold of saddles of a Morse-Smale flow on a closed manifold need not consist of finitely many connected components (or equivalently need not consist of finitely many abstract weak orbits). Therefore we study the finiteness of abstract weak orbit spaces of Morse(-Smale) flow on compact manifolds.

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arXiv2019

Existence and non-existence of length averages for foliations

Since the pioneering work of Ghys, Langevin and Walczak among others, it has been known that several methods of dynamical systems theory can be adopted to study of foliations. Our aim in this paper is to investigate complexity of foliations, by generalising existence problem of time averages in dynamical systems theory to foliations: It has recently been realised that a positive Lebesgue measure set of points without time averages only appears for complicated dynamical systems, such as dynamical systems with heteroclinic connections or homoclinic tangencies. In this paper, we introduce the concept of length averages to singular foliations, and attempt to collect interesting examples with/without length averages. In particular, we demonstrate that length averages exist everywhere for any codimension one $\mathscr{C}^1 $ orientable singular foliation without degenerate singularities on a compact surface under a mild condition on quasi-minimal sets of the foliation, which is in strong contrast to time averages of surface flows.

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arXiv2017

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

In this paper, we characterize several lower separation axioms $C_0, C_D$, $C_R$, $C_N$, $λ$-space, nested, $S_{YS}$, $S_{YY}$, $S_{YS}$, and $S_δ$ 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.

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arXiv2017

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.

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arXiv2017

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.

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arXiv2014

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 $\mathrm{P} \sqcup \mathop{\mathrm{Sing}}(v) = \{ x \in M \mid ω(x) \cup α(x) \subseteq \mathop{\mathrm{Sing}}(v) \}$, 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 $\overline{\mathrm{P}} = \overline{\mathrm{LD}} =\mathbb{T}^2$.

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arXiv2014

Genericity for non-wandering surface flows

Consider the set $χ^0_{\mathrm{nw}}$ 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 $χ^0_{\mathrm{nw}}$. 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 $\mathbb{K}^2$.

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arXiv2013

Recurrence, pointwise almost periodicity and orbit closure relation for flows and foliations

In this paper, we obtain a characterizations of the recurrence of a continuous vector field $w$ of a closed connected surface $M$ as follows. The following are equivalent: 1) $w$ is pointwise recurrent. 2)$w$ is pointwise almost periodic. 3) $w$ is minimal or pointwise periodic. Moreover, if $w$ is regular, then the following are equivalent: 1) $w$ is pointwise recurrent. 2)$w$ is minimal or the orbit space $M/w$ is either $[0,1]$, or $S^1$. 3) $R$ is closed (where $R := \{(x,y) \in M \times M \mid y \in \bar{O(x)} \}$ is the orbit closure relation). On the other hand, we show that the following are equivalent for a codimension one foliation $\mathcal{F}$ on a compact manifold: 1) $\mathcal{F}$ is pointwise almost periodic. 2) $\mathcal{F}$ is minimal or compact. 3) $\mathcal{F}$ is $R$-closed. Also we show that if a foliated space on a compact metrizable space is either minimal or is both compact and without infinite holonomy, then it is $R$-closed.

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arXiv2013

Extended orbit properties on surfaces

In this paper, we study "demi-caractéristique" and (Poisson) stability in the sense of Poincaré. Using the definitions á la Poincaré 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.".

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