We present a simple approach to questions of topological orbit equivalence for actions of countable groups on topological and smooth manifolds. For example, for any action of a countable group $Γ$ on a topological manifold where the fixed sets for any element are contained in codimension two submanifolds, every orbit equivalence is equivariant. Even in the presence of larger fixed sets, for actions preserving rigid geometric structures our results force sufficiently smooth orbit equivalences to be equivariant. For instance, if a countable group $Γ$ acts on ${\mathbb T}^n$ and the action is $C^1$ orbit equivalent to the standard action of $SL_n({\mathbb Z})$ on ${\mathbb T}^n$, then $Γ$ is isomorphic to $SL_n({\mathbb Z})$ and the actions are isomorphic. (The same result holds if we replace $SL_n(\mathbb Z)$ by a finite index subgroup.) We also show that preserving a geometric structure is an invariant of smooth orbit equivalence and give an application of our ideas to the theory of hyperbolic groups.
In the course of proving our theorems, we generalize a theorem of Sierpinski which says that a connected Hausdorff compact topological space is not the disjoint union of countably many closed sets. We prove a stronger statement that allows "small" intersections provided the space is locally connected. This implies that for any continuous action of a countable group $\G$ on a connected, locally connected, locally compact, Hausdorff topological space, where the fixed set of every element is "small", every orbit equivalence is equivariant.
We prove results about orbit closures and equidistribution for the SL(2,R) action on the moduli space of compact Riemann surfaces, which are analogous to the theory of unipotent flows. The proofs of the main theorems rely on the measure classification theorem of [EMi2] and a certain isolation property of closed SL(2,R) invariant manifolds developed in this paper.
We prove a classification theorem for transitive Anosov and pseudo-Anosov flows on closed 3-manifolds, up to orbit equivalence. In many cases, flows on a 3-manifold $M$ are completely determined by the set of free homotopy classes of their (unoriented) periodic orbits. The exceptional cases are flows with a special structure in their orbit space called a ``tree of scalloped regions"; in these cases the set of free homotopy classes of unoriented periodic orbits together with the additional data of a choice of sign for each $π_1(M)$-orbit of tree gives a complete invariant of orbit equivalence c…
An area-preserving diffeomorphism of an annulus has an "action function" which measures how the diffeomorphism distorts curves. The average value of the action function over the annulus is known as the Calabi invariant of the diffeomorphism, while the average value of the action function over a periodic orbit of the diffeomorphism is the mean action of the orbit. If an area-preserving annulus diffeomorphism is a rotation near the boundary of the annulus, and if its Calabi invariant is less than the maximum boundary value of the action function, then we show that the infimum of the mean action …
It is proven that the orbit-equivalence class of any essentially free probability-measure-preserving action of a free group $G$ is weakly dense in the space of actions of $G$.
In this paper we study topological rigidity of affine actions on compact connected metrizable abelian groups. We also classify one-parameter flows of translations upto orbit equivalence and discrete group actions by translations upto topological conjugacy.
The $f$-invariant is a notion of entropy for probability-measure-preserving actions of free groups. We show it is invariant under bounded orbit-equivalence.
In this note we extend the concept of topological stability from homeomorphisms to group actions on compact metric spaces, and prove that if an action of a finitely generated group is expansive and has the pseudo-orbit tracing property then it is topologically stable. This represents a group action version of the Walter's stability theorem. Moreover we give a class of group actions with topological stability or pseudo-orbit tracing property. On the other hand, we also provide a characterization of subshifts of finite type over finitely generated groups in term of pseudo-orbit tracing property.
In this paper, we consider semigroup actions of discrete countable semigroups on compact spaces by surjective local homeomorphisms. We introduce notions of continuous one-sided orbit equivalence and continuous orbit equivalence for semigroup actions, and characterize them in terms of the corresponding semi-groupoids and transformation groupoids respectively. Finally, we consider the case of semigroup actions by homeomorphisms and relate continuous orbit equivalence of semigroup actions to that of group actions.
Consider a smooth action of $\mathbb R^n$ on a connected manifold $M$, not necessarily compact, of dimension $m$ and rank $k$. Assume that $M$ is not a cylinder. Then there exists an orbit of the action of dimension $<(m+k)/2$. As a consequence, one shows that if there is a non-zero element of the ring of Pontrjagin classes of $M$ of degree $4\ell\geq 4$, then there exists an orbit of the action of dimension $\leq m-\ell-1$.
We construct a minimal topological action $\wp$ of a non-amenable group on a Cantor set $C$, which is non-uniquely ergodic and furthermore there exist ergodic invariant measures $μ_1$ and $μ_2$ such that $(\wp,C,μ_1)$ and $(\wp,C,μ_2)$ are not orbit equivalent measurable equivalence relations.
We discuss properties of orbits of (semi)group actions on locally compact groups G. In particular, we show that if a compactly generated locally compact abelian group acts distally on G then the closure of each of its orbits is a minimal closed invariant set (i.e. the action has [MOC]). We also show that for such an action distality is preserved if we go modulo any closed normal invariant subgroup and hence [MOC] is also preserved. We also show that any semigroup action on G has [MOC] if and only if the corresponding actions on a compact invariant metrizable subgroup K and on the quotient spac…
Since the work of Ornstein and Weiss in 1987 (J. Analyse Math. 48 (1987)) it has been understood that the natural category for classical ergodic theory would be probability measure preserving actions of discrete amenable groups. A conclusion of this work is that all such actions on nonatomic Lebesgue probability spaces were orbit equivalent. From this foundation two broad developements have been built. First, a full generalization of the various equivalence theories, including Ornstein's isomorphism theorem itself, exists. Second, one can use the characterization of discrete amenable actions a…
Drimbe and Vaes proved an orbit equivalence superrigidity theorem for left-right wreath product actions in the measurable setting. We establish the counterpart result in the topological setting for continuous orbit equivalence. This gives us minimal, topologically free actions that are continuous orbit equivalence superrigid. One main ingredient for the proof is to show continuous cocycle superrigidity for certain generalized full shifts, extending our previous result with Chung.
Consider a countable amenable group acting by homeomorphisms on a compact metrizable space. Chung and Li asked if expansiveness and positive entropy of the action imply existence of an off-diagonal asymptotic pair. For algebraic actions of polycyclic-by-finite groups, Chung and Li proved it does. We provide examples showing that Chung and Li's result is near-optimal in the sense that the conclusion fails for some non-algebraic action generated by a single homeomorphism, and for some algebraic actions of non-finitely generated abelian groups. On the other hand, we prove that every expansive act…
Let $Γ\curvearrowright (X,μ)$ be a measure preserving action of a countable group $Γ$ on a standard probability space $(X,μ)$. We prove that if the action $Γ\curvearrowright X$ is not profinite and satisfies a certain spectral gap condition, then there does not exist a countable-to-one Borel homomorphism from its orbit equivalence relation to the orbit equivalence relation of any modular action (i.e., an inverse limit of actions on countable sets). As a consequence, we show that if $Γ$ is a countable dense subgroup of a compact non-profinite group $G$ such that the left translation action $Γ\c…
A homeomorphism of the $2$-torus with Lefschetz number different from zero has a fixed point. We give a version of this result for nilpotent groups of diffeomorphisms. We prove that a nilpotent group of $2$-torus diffeomorphims has finite orbits when the group has some element with Lefschetz number different from zero.
We prove that if $Γ$ is a countable group without a subgroup isomorphic to $\mathbb{Z}^2$ that acts faithfully and minimally by orientation preserving homeomorphisms on the circle, then it has a free orbit. We give examples showing that this does not hold for actions by homeomorphisms of the line.
We will study torus actions on Cuntz--Krieger algebras trivially acting on its canonical maximal abelian $C^*$-subalgebras from the view points of continuous orbit equivalence of one-sided topological Markov shifts and flow equivalence of two-sided topological Markov shifts.
It is shown that the restriction of the action of any group with finite orbit on the minimal sets of dendrites is equicontinuous. Consequently, we obtain that the action of any amenable group and Thompson group on dendrite restricted on minimal sets is equicontinuous. We further provide a class of non-amenable groups whose action on dendrites has finite orbit. We extend also some of our results to dendron. We further give a characterization of the set of invariant probability measures and its extreme points.
In this paper we introduce the notion of orbit equivalence for semigroup actions and the concept of generalized linear control system on smooth manifold. The main goal is to prove that, under certain conditions, the semigroup system of a generalized linear control system on a smooth manifold is orbit equivalent to the semigroup system of a linear control system on a homogeneous space.
In this note we show that for any proper action of a Banach--Lie group $G$ on a Banach manifold $M$, the corresponding tangent maps $\g \to T_x(M)$ have closed range for each $x \in M$, i.e., the tangent spaces of the orbits are closed. As a consequence, for each free proper action on a Hilbert manifold, the quotient $M/G$ carries a natural manifold structure.
We give an elementary proof for Lewis Bowen's theorem saying that two Bernoulli actions of two free groups, each having arbitrary base probability spaces, are stably orbit equivalent. Our methods also show that for all compact groups K and every free product Γof n infinite amenable groups, the factor K^Γ/K of the Bernoulli action of Γon K^Γ by the diagonal action of K, is isomorphic with a Bernoulli action of Γ.
Quasi-invariant measures for non-discrete groups of diffeomorphisms containing a Morse-Smale dynamics are studied. The assumption concerning the presence of a Morse-Smale dynamics allows us to extend to higher dimensions a number of recently established results for non-discrete groups acting on the circle. These results are also applied to show that, for many groups as above, every continuous orbit equivalence must coincide almost everywhere with a diffeomorphism of the corresponding manifold.
Given a countable amenable group G and 0 < L < 1, we give an elementary construction of a type-III:L Bernoulli group action. In the case where G is the integers, we show that our nonsingular Bernoulli shifts have independent and identically distributed factors.
In this work, we study minimal equicontinuous actions which are locally quasi-analytic. The first main result shows that for minimal equicontinuous actions which are locally quasi-analytic, continuous orbit equivalence of the actions implies return equivalence. This generalizes results of Cortez and Medynets, and of Li. The second main result is that if G is a finitely-generated, virtually nilpotent group, then every minimal equicontinuous action by G is locally quasi-analytic. As an application, we show that the homeomorphism type of a nil-solenoid is determined by the virtual topological ful…
The following result is proven. Let $G_1 \cc^{T_1} (X_1,μ_1)$ and $G_2 \cc^{T_2} (X_2,μ_2)$ be orbit-equivalent, essentially free, probability measure preserving actions of countable groups $G_1$ and $G_2$. Let $H$ be any countable group. For $i=1,2$, let $Γ_i = G_i *H$ be the free product. Then the actions of $Γ_1$ and $Γ_2$ coinduced from $T_1$ and $T_2$ are orbit-equivalent. As an application, it is shown that if $Γ$ is a free group, then all nontrivial Bernoulli shifts over $Γ$ are orbit-equivalent.
We prove that the solid ergodicity property is stable with respect to taking coinduction for a fairly large class of coinduced action. More precisely, assume that $Σ<Γ$ are countable groups such that $gΣg^{-1}\cap Σ$ is finite for any $g\inΓ\setminusΣ$. Then any measure preserving action $Σ\curvearrowright X_0$ gives rise to a solidly ergodic equivalence relation if and only if the equivalence relation of the associated coinduced action $Γ\curvearrowright X$ is solidly ergodic. We also obtain orbit equivalence rigidity for such actions by showing that the orbit equivalence relation of a rigid …
Explicit construction of orbit equivalence for rank-one actions
Nicolaus Copernicus University（尼古拉·哥白尼大学）；B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine（乌克兰国家科学院 B.维尔金低温物理与工程研究所）；Imperial College London（伦敦帝国学院）
The main theme of this paper is the connection between the existence of infinitely many periodic orbits for a Hamiltonian system and the behavior of its action or index spectrum under iterations. We use the action and index spectra to show that any Hamiltonian diffeomorphism of a closed, rational manifold with zero first Chern class has infinitely many periodic orbits and that, for a general rational manifold, the number of geometrically distinct periodic orbits is bounded from below by the ratio of the minimal Chern number and half of the dimension. These generalizations of the Conley conject…
We show that every minimal, free action of the group Z^2 on the Cantor set is orbit equivalent to an AF-relation. As a consequence, this extends the classification of minimal systems on the Cantor set up to orbit equivalence to include AF-relations, Z-actions and Z^2-actions.
We prove that for any two continuous minimal (topologically free) actions of the infinite dihedral group on an infinite compact Hausdorff space, they are continuously orbit equivalent only if they are conjugate. We also show the above fails if we replace the infinite dihedral group with certain other virtually cyclic groups, e.g. the direct product of the integer group with any non-abelian finite simple group.
Let $Γ_1,\dots,Γ_n$ be hyperbolic, property (T) groups, for some $n\ge 1$. We prove that if a product $Γ_1\times\dots\timesΓ_n \curvearrowright X_1\times\dots\times X_n$ of measure preserving actions is stably orbit equivalent to a measure preserving action $Λ\curvearrowright Y$, then $Λ\curvearrowright Y$ is induced from an action $Λ_0\curvearrowright Y_0$ such that there exists a direct product decomposition $Λ_0=Λ_1\times\dots\timesΛ_n$ into $n$ infinite groups. Moreover, there exists a measure preserving action $Λ_i\curvearrowright Y_i$ that is stably orbit equivalent to $Γ_i\curvearrowrig…
Continuous actions of topological groups on compact Hausdorff spaces $X$ are investigated which induce almost periodic functions in the corresponding commutative C*-algebra. The unique invariant mean on the group resulting from averaging allows to derive a C*-valued inner product and a Hilbert C*-module which serve as an environment to describe characteristics of the group action. For uniformly continuous, Lyapunov stable actions the derived invariant mean $M(ϕ_x)$ is continuous on $X$ for any element $ϕ\in C(X)$, and the induced C*-valued inner product corresponds to a conditional expectation…
We consider a group $G$ acting on a local dendrite $X$ (in particular on a graph). We give a full characterization of minimal sets of $G$ by showing that any minimal set $M$ of $G$ (whenever $X$ is different from a dendrite) is either a finite orbit, or a Cantor set, or a circle. If $X$ is a graph different from a circle, such a minimal $M$ is a finite orbit. These results extend those of the authors for group actions on dendrites. On the other hand, we show that, for any group $G$ acting on a local dendrite $X$ different from a circle, the following properties are equivalent: (1) ($G, X$) is …
A classical theorem due to Wadsley implies that, on a connected contact manifold all of whose Reeb orbits are closed, there is a common period for the Reeb orbits. In this paper we show that, for any Reeb flow on a closed connected 3-manifold, the following conditions are actually equivalent: (1) every Reeb orbit is closed; (2) all closed Reeb orbits have a common period; (3) the action spectrum has rank 1. We also show that, on a fixed closed connected 3-manifold, a contact form with an action spectrum of rank 1 is determined (up to pull-back by diffeomorphisms) by the set of minimal periods …
We study equivalence relations that arise from translation actions $Γ\curvearrowright G$ which are associated to dense embeddings $Γ<G$ of countable groups into second countable locally compact groups. Assuming that $G$ is simply connected and the action $Γ\curvearrowright G$ is strongly ergodic, we prove that $Γ\curvearrowright G$ is orbit equivalent to another such translation action $Λ\curvearrowright H$ if and only if there exists an isomorphism $δ:G\rightarrow H$ such that $δ(Γ)=Λ$. If $G$ is moreover a real algebraic group, then we establish analogous rigidity results for the translation…
We take the first steps towards a better understanding of continuous orbit equivalence, i.e., topological orbit equivalence with continuous cocycles. First, we characterise continuous orbit equivalence in terms of isomorphisms of C*-crossed products preserving Cartan subalgebras. This is the topological analogue of the classical result by Singer and Feldman-Moore in the measurable setting. Secondly, we turn to continuous orbit equivalence rigidity, i.e., the question whether for certain classes of topological dynamical systems, continuous orbit equivalence implies conjugacy. We show that this …
By the work of Brodzki-Niblo-Nowak-Wright and Monod, topological amenability of a continuous group action can be characterized using uniformly finite homology groups or bounded cohomology groups associated to this action. We show that (certain variations of) these groups are invariants for topologically free actions under continuous orbit equivalence.
We say that two free probability-measure-preserving actions of countable groups are Shannon orbit equivalent if there is an orbit equivalence between them whose associated cocycle partitions have finite Shannon entropy. We show that if the acting groups are sofic and each has a w-normal amenable subgroup which is neither locally finite nor virtually cyclic then Shannon orbit equivalence implies that the actions have the same maximum sofic entropy. This extends a result of Austin beyond the finitely generated amenable setting and has the consequence that two Bernoulli actions of a group with th…
Multiplicity results for solutions of various boundary value problems are known for dynamical systems on compact configuration manifolds, given by Lagrangians or Hamiltonians which have quadratic growth in the velocities or in the momenta. Such results are based on the richness of the topology of the space of curves satisfying the given boundary conditions. In this note we show how these results can be extended to the classical setting of Tonelli Lagrangians (Lagrangians which are $C^2$-convex and superlinear in the velocities), or to Hamiltonians which are superlinear in the momenta and have …
We show in this work that the restricted three-body problem is in general not of contact type and zero is the energy value where the contact property breaks down. More explicitly, sequences of generating orbits with increasingly negative action and energies between $-\sqrt{2}$ and zero are constructed. Using results from Bolotin and MacKay, it is shown that these generating orbits extend to periodic solutions of the restricted three-body problem for small mass ratios and the action remains within a small neighbourhood. These orbits obstruct the existence of contact structures for energy level …
We show that every parabolic orbit of a two-degree of freedom integrable system admits a $C^\infty$-smooth Hamiltonian circle action, which is persistent under small integrable $C^\infty$ perturbations. We deduce from this result the structural stability of parabolic orbits and show that they are all smoothly equivalent (in the non-symplectic sense) to a standard model. Our proof is based on showing that every symplectomorphism of a neighbourhood of a parabolic point preserving the integrals of motion is Hamiltonian whose generating function is smooth and constant on the connected components o…
We consider the classical problem of area-preserving maps on annulus $\mathbb{A} = S^1 \times [0, 1]$ . Let $\mathcal{M}_f$ be the set of all invariant probability measures of an area-preserving, orientation preserving diffeomorphism $f$ on $\mathbb{A}$. Given any $μ_1$ and $μ_2$ in $\mathcal{M}_f$, Franks \cite{Franks1988}\cite{Franks1992}, generalizing Poincaré's last geometric theorem (Birkhoff \cite{Birkhoff1913}), showed that if their rotation numbers are different, then $f$ has infinitely many periodic orbits. In this paper, we show that if $μ_1$ and $μ_2$ have different actions, even if…
We study equivalence relations $\mathcal R(Γ\curvearrowright G)$ that arise from left translation actions of countable groups on their profinite completions. Under the assumption that the action $Γ\curvearrowright G$ is free and has spectral gap, we describe precisely when $\mathcal R(Γ\curvearrowright G)$ is orbit equivalent or Borel reducible to another such equivalence relation $\mathcal R(Λ\curvearrowright H)$. As a consequence, we provide explicit uncountable families of free ergodic probability measure preserving (p.m.p.) profinite actions of $SL_2(\mathbb Z)$ and its non-amenable subgro…
Let $X$ be a locally compact zero-dimensional space, let $S$ be an equicontinuous set of homeomorphisms such that $1 \in S = S^{-1}$, and suppose that $\overline{Gx}$ is compact for each $x \in X$, where $G = \langle S \rangle$. We show in this setting that a number of conditions are equivalent: (a) $G$ acts minimally on the closure of each orbit; (b) the orbit closure relation is closed; (c) for every compact open subset $U$ of $X$, there is $F \subseteq G$ finite such that $\bigcap_{g \in F}g(U)$ is $G$-invariant. All of these are equivalent to a notion of recurrence, which is a variation on…
We show that the topological rank of an orbit full group generated by an ergodic, probability measure-preserving free action of a non-discrete unimodular locally compact Polish group is two. For this, we use the existence of a cross section and show that for a locally compact Polish group, the full group generated by any dense subgroup is dense in the orbit full group of the action of the group. We prove that the orbit full group of a free action of a locally compact Polish group is extremely amenable if and only if the acting group is amenable, using the fact that the full group generates the…
We study Hamiltonian diffeomorphisms of closed symplectic manifolds with non-contractible periodic orbits. In a variety of settings, we show that the presence of one non-contractible periodic orbit of a Hamiltonian diffeomorphism of a closed toroidally monotone or toroidally negative monotone symplectic manifold implies the existence of infinitely many non-contractible periodic orbits in a specific collection of free homotopy classes. The main new ingredient in the proofs of these results is a filtration of Floer homology by the so-called augmented action. This action is independent of capping…
We show that in the neighborhood of each ``finite type'' singular orbit of a real analytic integrable dynamical system (hamiltonian or not) there is a real analytic torus action which preserves the system and which is transitive on this orbit. We also show that the local automorphism group of the system near such an orbit is
In this paper, we study orbit closures under diagonal torus actions. We show that if $(x,y)\in\mathbb{T}^2$ is not contained in any rational lines, then its orbit under the $\times 2, \times 3, \times 5$ actions is dense in $\mathbb{T}^2.$
The aim of this paper is to study the motion of $2+n$-body problem where two equal masses are assumed to be fixed. We assume that the value of each fixed mass is equal to $M>0$ and the remaining $n$ moving particles have equal masses $m>0$. According to Newton's second law and the universal gravitation law, the $n$ particles move under the interaction of each other and the affection of the two fixed particles. Also, this motion has a natural variational structure. Under the simple choreography constraint, we show that the Lagrangian action functional attains its absolute minimum on a uniform c…
