Orientation-preserving Homeomorphism Research Articles

Braids and linked periodic orbits of disc homeomorphisms

This paper concerns on linked periodic orbits of orientation-preserving homeomorphisms of the 2-disc in the sense of Gambaudo. We interpret the linking of periodic orbits by using their induced braids. Then based on the forcing relation of braids, we present a method for finding periodic orbits which are linked with a given one. In particular, for 3-periodic orbits of pseudo-Anosov braid types, we provide new examples of linked orbits of periods at most 4.

Bounded and unbounded cohomology of homeomorphism and diffeomorphism groups

We determine the bounded cohomology of the group of homeomorphisms of certain low-dimensional manifolds. In particular, for the group of orientation-preserving homeomorphisms of the circle and of the closed 2-disc, it is isomorphic to the polynomial ring generated by the bounded Euler class. These seem to be the first examples of groups for which the entire bounded cohomology can be described without being trivial. We further prove that the $$C^r$$ -diffeomorphisms groups of the circle and of the closed 2-disc have the same bounded cohomology as their homeomorphism groups, so that both differ from the ordinary cohomology of $$C^r$$ -diffeomorphisms when $$r>1$$ . Finally, we determine the low-dimensional bounded cohomology of homeo- and diffeomorphism of the spheres $$S^n$$ and of certain 3-manifolds. In particular, we answer a question of Ghys by showing that the Euler class in $$H^4(\textrm{Homeo}_\circ (S^3))$$ is unbounded.

On a Classification of Periodic Maps on the 2-Torus

In this paper, following J. Nielsen, we introduce a complete characteristic of orientation-preserving periodic maps on the two-dimensional torus. All admissible complete characteristics were found and realized. In particular, each of the classes of orientation-preserving periodic homeomorphisms on the 2-torus that are nonhomotopic to the identity is realized by an algebraic automorphism. Moreover, it is shown that the number of such classes is finite. According to V. Z. Grines and A. Bezdenezhnykh, any gradient-like orientation-preserving diffeomorphism of an orientable surface is represented as a superposition of the time-1 map of a gradient-like flow and some periodic homeomorphism. Thus, the results of this work are directly related to the complete topological classification of gradient-like diffeomorphisms on surfaces.

Two‐dimensional metric spheres from gluing hemispheres

We study metric spheres ( Z , d Z ) $(Z, d_{Z} )$ obtained by gluing two hemispheres of S 2 $\mathbb {S}^{2}$ along an orientation-preserving homeomorphism g : S 1 → S 1 $g \colon \mathbb {S}^{1} \rightarrow \mathbb {S}^{1}$ , where d Z $d_{Z}$ is the canonical distance that is locally isometric to S 2 $\mathbb {S}^{2}$ off the seam. We show that if ( Z , d Z ) $(Z, d_{Z} )$ is quasiconformally equivalent to S 2 $\mathbb {S}^{2}$ , in the geometric sense, then g $g$ is a welding homeomorphism with conformally removable welding curves. We also show that g $g$ is bi-Lipschitz if and only if ( Z , d Z ) $(Z, d_{Z} )$ has a 1-quasiconformal parametrization whose Jacobian is comparable to the Jacobian of a quasiconformal mapping h : S 2 → S 2 $h \colon \mathbb {S}^{2} \rightarrow \mathbb {S}^{2}$ . Furthermore, we show that if g − 1 $g^{-1}$ is absolutely continuous and g $g$ admits a homeomorphic extension with exponentially integrable distortion, then ( Z , d Z ) $(Z, d_{Z} )$ is quasiconformally equivalent to S 2 $\mathbb {S}^{2}$ .

Angular Values of Nonautonomous and Random Linear Dynamical Systems: Part I---Fundamentals

We introduce the notion of angular values for deterministic linear difference equations and ran- dom linear cocycles. We measure the principal angles between subspaces of fixed dimension as they evolve under nonautonomous or random linear dynamics. The focus is on long-term averages of these principal angles, which we call angular values: we demonstrate relationships between different types of angular values and prove their existence for random dynamical systems. For one-dimensional subspaces in two-dimensional systems our angular values agree with the classical theory of rotation numbers for orientation-preserving circle homeomorphisms if the matrix has positive determinant and does not rotate vectors by more than $\frac{\pi}{2}$. Because our notion of angular values ignores orientation by looking at subspaces rather than vectors, our results apply to dynamical systems of any dimension and to subspaces of arbitrary dimension. The second part of the paper delves deeper into the theory of the autonomous case. We explore the relation to (generalized) eigenspaces, provide some explicit formulas for angular values, and set up a general numerical algorithm for computing angular values via Schur decompositions.

Direct decompositions of groups of piecewise linear homeomorphisms of the unit interval

We study subgroups of the group [Formula: see text] of piecewise linear orientation-preserving homeomorphisms of the unit interval [Formula: see text] that are differentiable everywhere except at finitely many real numbers, under the operation of composition. We provide a criterion for any two subgroups of [Formula: see text] which are direct products of finitely many indecomposable non-commutative groups to be non-isomorphic. As its application, we give a necessary and sufficient condition for any two subgroups of the R. Thompson group [Formula: see text] that are stabilizers of finite sets of numbers in the interval [Formula: see text] to be isomorphic, thus solving a problem by G. Golan and M. Sapir. We also show that if two stabilizers are isomorphic, then they are conjugate inside a certain group [Formula: see text].

On non-hyperbolic algebraic automorphisms of a two-dimensional torus

Abstract. This paper contains a complete classification of algebraic non-hyperbolic automorphisms of a two-dimensional torus, announced by S. Batterson in 1979. Such automorphisms include all periodic automorphisms. Their classification is directly related to the topological classification of gradient-like diffeomorphisms of surfaces, since according to the results of V. Z. Grines and A.N. Bezdenezhykh, any gradient like orientation-preserving diffeomorphism of an orientable surface is represented as a superposition of the time-1 map of a gradient-like flow and some periodic homeomorphism. J. Nielsen found necessary and sufficient conditions for the topological conjugacy of orientation-preserving periodic homeomorphisms of orientable surfaces by means of orientation-preserving homeomorphisms. The results of this work allow us to completely solve the problem of realization all classes of topological conjugacy of periodic maps that are not homotopic to the identity in the case of a torus. Particularly, it follows from the present paper and the work of that if the surface is a two-dimensional torus, then there are exactly seven such classes, each of which is represented by algebraic automorphism of a two-dimensional torus induced by some periodic matrix.

Determining solubility for finitely generated groups of PL homeomorphisms

The set of finitely generated subgroups of the group P L + ( I ) PL_+(I) of orientation-preserving piecewise-linear homeomorphisms of the unit interval includes many important groups, most notably R. Thompson’s group F F . Here, we show that every finitely generated subgroup G > P L + ( I ) G>PL_+(I) is either soluble, or contains an embedded copy of the finitely generated, non-soluble Brin-Navas group B B , affirming a conjecture of the first author from 2009. In the case that G G is soluble, we show the derived length of G G is bounded above by the number of breakpoints of any finite set of generators. We specify a set of ‘computable’ subgroups of P L + ( I ) PL_+(I) (which includes R. Thompson’s group F F ) and give an algorithm which determines whether or not a given finite subset X X of such a computable group generates a soluble group. When the group is soluble, the algorithm also determines the derived length of ⟨ X ⟩ \langle X\rangle . Finally, we give a solution of the membership problem for a particular family of finitely generated soluble subgroups of any computable subgroup of P L + ( I ) PL_+(I) .

On C*-algebras of singular integral operators with PQC coefficients and shifts with fixed points

A Fredholm representation on a Hilbert space, whose kernel coincides with the ideal of compact operators, is constructed for the -algebra generated by all multiplication operators by piecewise quasicontinuous (PQC) functions, by the Cauchy singular integral operator and by the unitary weighted shift operators , acting on the space over the unit circle . Here G denotes a discrete amenable group of orientation-preserving piecewise smooth homeomorphisms with finite sets of discontinuities for their derivatives , which acts topologically freely on , where is the interior of the nonempty closed set composed by all common fixed points for all shifts , with boundary of zero Lebesgue measure. A Fredholm symbol calculus for the -algebra is constructed and a Fredholm criterion for the operators is established.

The BNSR-invariants of the Stein group 𝐹2,3

Abstract The Stein group F 2 , 3 F_{2,3} is the group of orientation-preserving piecewise linear homeomorphisms of the unit interval with slopes of the form 2 p ⁢ 3 q 2^{p}3^{q} ( p , q ∈ Z p,q\in\mathbb{Z} ) and breakpoints in Z ⁢ [ 1 6 ] \mathbb{Z}[\frac{1}{6}] . This is a natural relative of Thompson’s group 𝐹. In this paper, we compute the Bieri–Neumann–Strebel–Renz (BNSR) invariants Σ m ⁢ ( F 2 , 3 ) \Sigma^{m}(F_{2,3}) of the Stein group for all m ∈ N m\in\mathbb{N} . A consequence of our computation is that (as with 𝐹) every finitely presented normal subgroup of F 2 , 3 F_{2,3} is of type F ∞ \operatorname{F}_{\infty} . Another, more surprising, consequence is that (unlike 𝐹) the kernel of any map F 2 , 3 → Z F_{2,3}\to\mathbb{Z} is of type F ∞ \operatorname{F}_{\infty} , even though there exist maps F 2 , 3 → Z 2 F_{2,3}\to\mathbb{Z}^{2} whose kernels are not even finitely generated. In terms of BNSR-invariants, this means that every discrete character lies in Σ ∞ ⁢ ( F 2 , 3 ) \Sigma^{\infty}(F_{2,3}) , but there exist (non-discrete) characters that do not even lie in Σ 1 ⁢ ( F 2 , 3 ) \Sigma^{1}(F_{2,3}) . To the best of our knowledge, F 2 , 3 F_{2,3} is the first group whose BNSR-invariants are known exhibiting these properties.

Realization of homotopy classes of torus homeomorphisms by the simplest structurally stable diffeomorphisms

Abstract. According to Thurston’s classification, the set of homotopy classes of orientation-preserving homeomorphisms of orientable surfaces is split into four disjoint subsets. A homotopy class from each subset is characterized by the existence of a homeomorphism called Thurston’s canonical form, namely: a periodic homeomorphism, a reducible nonperiodic homeomorphism of algebraically finite order, a reducible homeomorphism that is not a homeomorphism of an algebraically finite order, and a pseudo-Anosov homeomorphism. Thurston’s canonical forms are not structurally stable diffeomorphisms. Therefore, the problem naturally arises of constructing the simplest (in a certain sense) structurally stable diffeomorphisms in each homotopy class. In this paper, the problem posed is solved for torus homeomorphisms. In each homotopy class, structurally stable representatives are analytically constructed, namely, a gradient-like diffeomorphism, a Morse-Smale diffeomorphism with an orientable heteroclinic, and an Anosov diffeomorphism, which is a particular case of a pseudo-Anosov diffeomorphism.

Enumeration of rooted 4-regular maps without planar loops

With the help of composition formula we obtain a very simple expression that enables us to find the generating function for rooted 4-regular maps without planar loops on an arbitrary orientable or non-orientable surface of a given genus g, assuming that the generating function for all rooted 4-regular maps on this surface is known. As an application of this result, we derive explicit expressions for the corresponding generating functions for maps on the sphere, the projective plane, the torus, and the Klein bottle. We also provide an explicit formula for the number of unlabelled 4-regular maps without planar loops on the sphere, counted up to orientation-preserving homeomorphisms.

On uniqueness of invariant measures for random walks on

AbstractWe consider random walks on the group of orientation-preserving homeomorphisms of the real line ${\mathbb R}$ . In particular, the fundamental question of uniqueness of an invariant measure of the generated process is raised. This problem was studied by Choquet and Deny [Sur l’équation de convolution $\mu = \mu * \sigma $ . C. R. Acad. Sci. Paris250 (1960), 799–801] in the context of random walks generated by translations of the line. Nowadays the answer is quite well understood in general settings of strongly contractive systems. Here we focus on a broader class of systems satisfying the conditions of recurrence, contraction and unbounded action. We prove that under these conditions the random process possesses a unique invariant Radon measure on ${\mathbb R}$ . Our work can be viewed as following on from Babillot et al [The random difference equation $X_n=A_n X_{n-1}+B_n$ in the critical case. Ann. Probab.25(1) (1997), 478–493] and Deroin et al [Symmetric random walk on $\mathrm {HOMEO}^{+}(\mathbb {R})$ . Ann. Probab.41(3B) (2013), 2066–2089].

Goeritz Groups of Bridge Decompositions

Abstract For a bridge decomposition of a link in the $3$-sphere, we define the Goeritz group to be the group of isotopy classes of orientation-preserving homeomorphisms of the $3$-sphere that preserve each of the bridge sphere and link setwise. After describing basic properties of this group, we discuss the asymptotic behavior of the minimal pseudo-Anosov entropies. We then give an application to the asymptotic behavior of the minimal entropies for the original Goeritz groups of Heegaard splittings of the $3$-sphere and the real projective space.

Determination of the Homotopy Type of a Morse – Smale Diffeomorphism on a 2-torus by Heteroclinic Intersection

According to the Nielsen – Thurston classification, the set of homotopy classes of orientation-preserving homeomorphisms of orientable surfaces is split into four disjoint subsets. Each subset consists of homotopy classes of homeomorphisms of one of the following types: $T_{1}$) periodic homeomorphism; $T_{2}$) reducible non-periodic homeomorphism of algebraically finite order; $T_{3}$) a reducible homeomorphism that is not a homeomorphism of algebraically finite order; $T_{4}$) pseudo-Anosov homeomorphism. It is known that the homotopic types of homeomorphisms of torus are $T_{1}$, $T_{2}$, $T_{4}$ only. Moreover, all representatives of the class $T_{4}$ have chaotic dynamics, while in each homotopy class of types $T_{1}$ and $T_{2}$ there are regular diffeomorphisms, in particular, Morse – Smale diffeomorphisms with a finite number of heteroclinic orbits. The author has found a criterion that allows one to uniquely determine the homotopy type of a Morse – Smale diffeomorphism with a finite number of heteroclinic orbits on a two-dimensional torus. For this, all heteroclinic domains of such a diffeomorphism are divided into trivial (contained in the disk) and non-trivial. It is proved that if the heteroclinic points of a Morse – Smale diffeomorphism are contained only in the trivial domains then such diffeomorphism has the homotopic type $T_{1}$. The orbit space of non-trivial heteroclinic domains consists of a finite number of two-dimensional tori, where the saddle separatrices participating in heteroclinic intersections are projected as transversally intersecting knots. That whether the Morse – Smale diffeomorphisms belong to types $T_{1}$ or $T_{2}$ is uniquely determined by the total intersection index of such knots.

On a class of topological conjugacy with a homothety

We consider a class H(Rn) of orientation-preserving homeomorphisms of Euclidean space Rn such that for any homeomorphism h∈H(Rn) and for any point x∈Rn a condition limn→+∞hn(x)→O holds, were O is the origin. It is proved that for any n≥1 an arbitrary homeomorphism h∈H(Rn) is topologically conjugated with the homothety an:Rn→Rn, given by an(x1,…,an)=(12x1,…,12xn). For a smooth case under the condition that all eigenvalues of the differential of the mapping h have absolute values smaller than one, this fact follows from the classical theory of dynamical systems. In the topological case for n∉{4,5} this fact is proven in several works of 20th century, but authors do not know any papers where it would be proven for n∈{4,5}. This paper fills this gap.

On the Non-vanishing of the Powers of the Euler Class for Mapping Class Groups

The mapping class group of an orientable closed surface with one marked point can be identified, by the Nielsen action, with a subgroup of the group of orientation-preserving homeomorphisms of the circle. This inclusion pulls back the “discrete universal Euler class” producing a non-zero class in the second integral cohomology of the mapping class group. In this largely expository note, we determine the non-vanishing behavior of the powers of this class. Our argument relies on restricting the cohomology classes to torsion subgroups of the mapping class group.

Ergodicity and central limit theorem for random interval homeomorphisms

The central limit theorem for Markov chains generated by iterated function systems consisting of orientation-preserving homeomorphisms of the interval is proved. We study also ergodicity of such systems.

Non-realizability of the pure braid group as area-preserving homeomorphisms

Let$\operatorname{Homeo}_{+}(D_{n}^{2})$be the group of orientation-preserving homeomorphisms of$D^{2}$fixing the boundary pointwise and$n$marked points as a set. The Nielsen realization problem for the braid group asks whether the natural projection$p_{n}:\operatorname{Homeo}_{+}(D_{n}^{2})\rightarrow B_{n}:=\unicode[STIX]{x1D70B}_{0}(\operatorname{Homeo}_{+}(D_{n}^{2}))$has a section over subgroups of$B_{n}$. All of the previous methods use either torsion or Thurston stability, which do not apply to the pure braid group$PB_{n}$, the subgroup of$B_{n}$that fixes$n$marked points pointwise. In this paper, we show that the pure braid group has no realization inside the area-preserving homeomorphisms using rotation numbers.

Invertibility Criteria in $${\varvec{C^*}}$$ C ∗ -algebras of Functional Operators with Shifts and $${\varvec{PQC}}$$ PQC Coefficients

Let G be an amenable discrete group of orientation-preserving piecewise smooth homeomorphisms $$g:\mathbb {T}\rightarrow \mathbb {T}$$ , with finite sets of discontinuities for their derivatives $$g'$$ , which acts topologically freely on $$\mathbb {T}{\setminus }\Lambda ^\circ $$ , where $$\Lambda ^\circ $$ is the interior of a nonempty closed set $$\Lambda \subset \mathbb {T}$$ composed by all common fixed points for all shifts $$g\in G$$ . Invertibility criteria are established for the operators in the $$C^*$$ -algebra $$\begin{aligned} {\mathcal {A}}:=\mathrm{alg}\,(PQC,U_G)\subset {\mathcal {B}}(L^2(\mathbb {T})) \end{aligned}$$ generated by all functional operators of the form $$\sum _{g\in F}a_gU_g$$ , where $$a_gI$$ are multiplication operators by piecewise quasicontinuous functions $$a_g\in PQC$$ on $$\mathbb {T}$$ , $$U_g:\varphi \mapsto |g'|^{1/2} (\varphi \circ g)$$ are unitary weighted shift operators on $$L^2(\mathbb {T})$$ , and F is any finite subset of the group G.