On the Centralizer and Conjugacy of Pseudo-Anosov Homeomorphisms

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.

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.

In the present paper axiom diffeomorphisms of closed 2-manifolds of genus whose nonwandering set contains perfect spaciously situated one-dimensional attractor are considered. It is shown that such diffeomorphisms are topologically semiconjugate to pseudo-Anosov homeomorphism with the same induced automorphism of fundamental group. The main result of the paper is the following. Two diffeomorphisms from the given class are topologically conjugate on attractors if and only if corresponding pseudo-Anosov homeomorphisms are topologically conjugate by means of homeomorphism that maps a certain subset of one pseudo-Anosov map onto the certain subset of the other pseudo-Anosov map.

S. Kh. Aranson, I. U. Bronstein, E. V. Zhuzhoma, and I.V. Nikolaev UDC 517.987.5; 517.933 Preface Foliations on surfaces are natural generalizations of flows (continuous-time dynamical systems) on sur- faces. According to the classical theory, in the neighborhood of a nonsingular point (i.e., a point different from the equilibrium state), the trajectories of a flow are a family of parallel straight lines. This is what served as a starting point for defining a foliation. Let M be a closed surface (a closed two-dimensional manifold). The foliation F on M with a set of singularities Sing (F) is the partitioning M - Sing (F) into nonintersecting curves (called fibers) which are locally homeomorphic to the family of straight lines. As to the set of singularities Sing (F), this set must be described separately for every class of fibers. In the sequel, unless otherwise specified, we shall consider the set Sing (F) to be finite. Foliations occupy an intermediate place between flows and arbitrary families of curves on surfaces. Foli- ations which can be embedded into a flow are said to be orientable; otherwise they are nonorientable. The origination of a qualitative theory of foliations goes back to the works by H. Kneser, G. Reeb, A. Haefliger, and S. P. Novikov. The theory of foliations attracted special attention early in the sixties in connection with the study of Y-flows and Y-diffeomorphisms introduced by D. V. Anosov. The technique of foliations allowed G. Franks to classify Y-diffeomorphisms of codimension one the nonwandering set of whose points coincides with the whole manifold. The application of a "surgical operation" to Y-diffeomorphisms of codimension one leads to nontrivial base sets of codimension one (attractors and repellers) the profound results in the study of whose geometry and topology belong to P. V. Plykin and V. Z. Grines. A more general approach was suggested by Ya. G. Sinai, who introduced a class of dynamical systems with two invariant transversal fibers. When we use this approach, we neglect such properties of the systems as the estimates of the contraction and extension of the fibers of invariant foliations, the everywhere density of the periodic trajectories in a nonwandering set, and others, and leave only those properties which are closely connected with the topology' of the manifolds. A new impetus to the study of foliations and homeomorphisms with invariant foliations was given by the works of W. Thurston, in which he complemented the homotopic classification of homeomorphisms of surfaces obtained by d. Nielson in 1920-30 and gave a new interpretation to it. The introduction by Thurston of the concept of pseudo-Anosov homeomorphism, which generalized the concept of the Anosov diffeomorphism, stimulated further investigations in this direction based on the study of the action of homeomorphisms in a fundamental group. Beginning in the 1980s, the geometry and topology of foliations with saddle-point singularities were studied by H. Rosenberg (the construction of labyrinths), G. Levitt (equivalence in the sense of Whitehead), G. Papandopoulos, K. Dantoni, and others. At the same time, many questions concerning the qualitative theory of foliations (the topological and smooth classifications, the structural stability, typicalness and so on) remained open. This review is devoted to the contemporary state of the qualitative theory of foliations with singularities on closed surfaces, tile main emphasis being placed on the development of the theory of Poincard-Bendixon Translated from ltogi Nauki i Tekhniki, Seriya Sovreme,l,mya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 32, Dynanlical Systems-5, 1996. 1072-3374/98/9003-2111520.00 9 Plenum Publishing Corporation 2111

We develop a thermodynamic formalism for a smooth realization of pseudo-Anosov surface homeomorphisms. In this realization, the singularities of the pseudo-Anosov map are assumed to be fixed, and the trajectories are slowed down so the differential is the identity at these points. Using Young towers, we prove existence and uniqueness of equilibrium states for geometric t-potentials. This family of equilibrium states includes a unique SRB measure and a measure of maximal entropy, the latter of which has exponential decay of correlations and the central limit theorem.

Soit f un homeomorphisme pseudo-Anosov. On etudie l'existence d'une constante n f telle que π 1 (M f n)±{1} pour tout n>n f

By proving precisely which singularity index lists arise from the pair of invariant foliations for a pseudo-Anosov surface homeomorphism, Masur and Smillie [24] determined a Teichmüller flow invariant stratification of the space of quadratic differentials. In this final paper of a three-paper series, we give a first step to an [Formula: see text] analog of the Masur–Smillie theorem. Since the ideal Whitehead graphs defined by Handel and Mosher [16] give a strictly finer invariant in the analogous [Formula: see text] setting, we determine which of the 21 connected, simplicial, five-vertex graphs are ideal Whitehead graphs of fully irreducible outer automorphisms in [Formula: see text]. The rank 2 case is actually a direct consequence of the work of Masur and Smillie, as all elements of [Formula: see text] are induced by surface homeomorphisms and the index list and ideal Whitehead graph for a surface homeomorphism give precisely the same data.

We consider homeomorphisms$f,h$generating a faithful$\mathit{BS}(1,n)$-action on a closed surface$S$, that is,$hfh^{-1}=f^{n}$for some$n\geq 2$. According to Guelman and Liousse [Actions of Baumslag–Solitar groups on surfaces.Discrete Contin. Dyn. Syst. A 5(2013), 1945–1964], after replacing$f$by a suitable iterate if necessary, we can assume that there exists a minimal set$\unicode[STIX]{x1D6EC}$of the action, included in$\text{Fix}(f)$. Here, we suppose that$f$and$h$are$C^{1}$in a neighborhood of$\unicode[STIX]{x1D6EC}$and any point$x\in \unicode[STIX]{x1D6EC}$admits an$h$-unstable manifold$W^{u}(x)$. Using Bonatti’s techniques, we prove that either there exists an integer$N$such that$W^{u}(x)$is included in$\text{Fix}(f^{N})$or there is a lower bound for the norm of the differential of$h$depending only on$n$and the Riemannian metric on $S$. Combining the last statement with a result of Alonso, Guelman and Xavier [Actions of solvable Baumslag–Solitar groups on surfaces with (pseudo)-Anosov elements.Discrete Contin. Dyn. Syst.to appear], we show that any faithful action of$\mathit{BS}(1,n)$on$S$with$h$a pseudo-Anosov homeomorphism has a finite orbit containing singularities of $h$; moreover, if$f$is isotopic to the identity, it is entirely contained in the singular set of $h$. As a consequence, there is no faithful$C^{1}$-action of$\mathit{BS}(1,n)$on the torus with$h$Anosov.

An infinite family of generalized pseudo-Anosov homeomorphisms of the sphere\nS is constructed, and their invariant foliations and singular orbits are\ndescribed explicitly by means of generalized train tracks. The complex strucure\ninduced by the invariant foliations is described, and is shown to make S into a\ncomplex sphere. The generalized pseudo-Anosovs thus become quasiconformal\nautomorphisms of the Riemann sphere, providing a complexification of the\nunimodal family which differs from that of the Fatou/Julia theory.\n

In their precedent work, the authors constructed closed oriented hyperbolic surfaces with pseudo-Anosov homeomorphisms from certain class of integral matrices. In this paper, we present a very simple algorithm to compute the Teichmuller polynomial corresponding to those surface homeomorphisms by first constructing an invariant track whose first homology group can be naturally identified with the first homology group of the surface, and computing its Alexander polynomial.

There are two objects naturally associated with a braid β ∈ Bn of pseudo-Anosov type: a (relative) pseudo-Anosov homeomorphism ϕ β : S 2 → S 2 ; and the finite volume complete hyperbolic structure on the 3-manifold M β obtained by excising the braid closure of β, together with its braid axis, from S 3 .We show the disconnect between these objects, by exhibiting a family of braids {βq : q ∈ Q ∩ (0, 1/3]} with the properties that: on the one hand, there is a fixed homeomorphism ϕ0 : S 2 → S 2 to which the (suitably normalized) homeomorphisms ϕ βq converge as q → 0; while on the other hand, there are infinitely many distinct hyperbolic 3-manifolds which arise as geometric limits of the form lim k→∞ M βq k , for sequences q k → 0.

In this note, we show that given a closed, orientable genus-g surface Sg, any hyperbolic toral automorphism has a positive power which induces a quadratic, orientable pseudo-Anosov homeomorphism on Sg. To show this, we lift Anosov toral automorphisms through a ramified topological covering and present the lifted homeomorphism via a standard set of Lickorish twists. This construction provides a general method of producing pseudo-Anosov maps of closed surfaces with predetermined orientable foliations and quadratic dilatation. Since these lifted automorphisms have orientable foliations, this construction is a sort of converse to that of Franks and Rykken [Trans. Amer. Math. Soc. 1999], who established that one can associate to a quadratic pseudo-Anosov homeomorphism with oriented unstable foliation a hyperbolic toral automorphism.

We describe a circle of ideas relating the dynamics of 2-dimensional homeomorphisms to that of 1-dimensional endomorphisms. This is used to introduce a new class of maps generalizing that of Thurston's pseudo-Anosov homeomorphisms.

We give explicit pseudo-Anosov homeomorphisms with vanishingSah-Arnoux-Fathi invariant. Any translation surface whose Veech groupis commensurable to any of a large class of triangle groups is shownto have an affine pseudo-Anosov homeomorphism of this type. We alsoapply a reduction to finite triangle groups and thereby show theexistence of nonparabolic elements in the periodic field of certaintranslation surfaces.

By proving precisely which singularity index lists arise from the pair of invariant foliations for a pseudo-Anosov surface homeomorphism, Masur and Smillie [MS93] determined a Teichmüller flow invariant stratification of the space of quadratic differentials. In this paper we determine an analog to the theorem forOut(F3). That is, we determine which index lists permitted by the [GJLL98] index sum inequality are achieved by ageometric fully irreducible outer automorphisms of the rank-3 free group.