Let S be a connected orientable surface with finitely many punctures, finitely many boundary components, and genus at least 6. Then any C^1 action of the mapping class group of S on the circle is trivial.
The techniques used in the proof of this result permit us to show that products of Kazhdan groups and certain lattices cannot have C^1 faithful actions on the circle. We also prove that for n > 5, any C^1 action of Aut(F_n) or Out(F_n) on the circle factors through an action of Z/2Z.
In this paper we establish $C^2$ a-priori bounds for the scaling ratios of critical circle mappings in a form that gives also a compactness property for the renormalization operator.
We prove that any two real-analytic critical circle maps with cubic critical point and the same irrational rotation number of bounded type are $C^{1+α}$ conjugate for some $α>0$.
We show that if $G_1$ and $G_2$ are non-solvable groups, then no $C^{1,τ}$ action of $(G_1\times G_2)*\mathbb{Z}$ on $S^1$ is faithful for $τ>0$. As a corollary, if $S$ is an orientable surface of complexity at least three then the critical regularity of an arbitrary finite index subgroup of the mapping class group $\mathrm{Mod}(S)$ with respect to the circle is at most one, thus strengthening a result of the first two authors with Baik.
Given a uniform foliation by Gromov hyperbolic leaves on a $3$-manifold, we show that the action of the fundamental group on the universal circle is minimal and transitive on pairs of different points. We also prove two other results: we prove that general uniform Reebless foliations are $\mathbb{R}$-covered and we give a new description of the universal circle of $\mathbb{R}$-covered foliations with Gromov hyperbolic leaves in terms of the JSJ decomposition of $M$.
We prove that two $C^r$ critical circle maps with the same rotation number of bounded type are $C^{1+α}$ conjugate for some $α>0$ provided their successive renormalizations converge together at an exponential rate in the $C^0$ sense. The number $α$ depends only on the rate of convergence. We also give examples of $C^\infty$ critical circle maps with the same rotation number that are not $C^{1+β}$ conjugate for any $β>0$.
We study random elements of subgroups (and cosets) of the mapping class group of a closed hyperbolic surface, in part through the properties of their mapping tori. In particular, we study the distribution of the homology of the mapping torus (with rational, integer, and finite field coefficients, the hyperbolic volume (whenever the manifold is hyperbolic), the dilatation of the monodromy, the injectivity radius, and the bottom eigenvalue of the Laplacian on these mapping tori. We also study mapping tori of punctured surface bundles, and various invariants of their Dehn fillings. We also study …
Let $Σ_{g}$ be a closed, connected, and oriented surface of genus $g \geq 24$ and let $Γ$ be a finite index subgroup of the mapping class group $Mod(Σ_{g})$ that contains the Torelli group $\mathcal{I}(Σ_g)$. Then any orientation preserving $C^1$ action of $Γ$ on the circle cannot be faithful. We also show that if $Γ$ is a finite index subgroup of $Aut(F_n)$, when $n \geq 8$, that contains the subgroup of IA-automorphisms, then any orientation preserving $C^1$ action of $Γ$ on the circle cannot be faithful. Similarly, if $Γ$ is a finite index subgroup of $Out(F_n)$, when $n \geq 8$, that conta…
Let (M, F) be a compact codimension-one foliated manifold whose leaves are equipped with Riemannian metrics, and consider continuous functions on M that are harmonic along the leaves of F . If every such function is constant on leaves we say that (M, F) has the Liouville property. Our main result is that codimension-one foliated bundles over compact negatively curved manifolds satisfy the Liouville property. Related results for R-covered foliations, as well as for discrete group actions and discrete harmonic functions, are also established.
We characterize the semi-conjugacy class of a Fuchsian action of the modular group on the circle in terms of rotation numbers of two standard generators and that of their product. We also show that among lifts of a Fuchsian action of the modular group, only 5-fold lift admits a similar characterization. These results indicate similarity and difference between rotation number and linear character.
In this article, we describe all the group morphisms from the group of orientation-preserving homeomorphisms of the circle to the group of homeomorphisms of the annulus or of the torus.
We survey rigidity results for groups acting on the circle in various settings, from local to global and $C^0$ to smooth. Our primary focus is on actions of surface groups, with the aim of introducing the reader to recent developments and new tools to study groups acting by homeomorphisms.
We study topological properties of semi-group actions on the circle by orientation-preserving homeomorhisms. We prove that a generic action either possesses a forward-invariant interval-domain (i.e. a finite union of disjoint circle arcs), or is two-sided minimal (and, moreover, by a small perturbation one can create a map with arbitrary rotation number in the semigroup). For the minimal case, we also study conditions required for global synchronization.
This paper provides a canonical compactification of the plane ${\mathbb R}^2$ by adding a circle at infinity associated to a countable family of singular foliations or laminations (under some hypotheses), generalizing an idea by Mather \cite{Ma}. Moreover any homeomorphism of ${\mathbb R}^2 $ preserving the foliations extends on the circle at infinity. Then this paper provides conditions ensuring the minimality of the action on the circle at infinity induced by an action on ${\mathbb R}^2 $ preserving one foliation or two transverse foliations. In particular the action on the circle at infinit…
We show that for a finitely generated group of $C^2$ circle diffeomorphisms, the entropy of the action equals the entropy of the restriction of the action to the non-wandering set.
In this short and elementary note, we study some ergodic optimization problems for circle expanding maps. We first make an observation that if a function is not far from being convex, then its calibrated sub-actions are closer to convex functions in certain effective way. As an application of this simple observation, for circle doubling map, we generalize a result of Bousch saying that translations of the cosine function are uniquely optimized by Sturmian measures. Our argument follows the mainline of Bousch's original proof, while the technical part is simplified by the observation mentioned …
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 consider finitely generated groups of real-analytic circle diffeomorphisms. We show that if such a group admits an exceptional minimal set (i.e., a minimal invariant Cantor set), then its Lebesgue measure is zero; moreover, there are only finitely many orbits of connected components of its complement. For the case of minimal actions, we show that if the underlying group is (algebraically) free, then the action is ergodic with respect to the Lebesgue measure. This provides first answers to questions due to É. Ghys, G. Hector and D. Sullivan.
This work is devoted to the study of minimal, smooth actions of finitely generated groups on the circle. We provide a sufficient condition for such an action to be ergodic (with respect to the Lebesgue measure), and we illustrate this condition by studying two relevant examples. Under an analogous hypothesis, we also deal with the problem of the zero Lebesgue measure for exceptional minimal sets. This hypothesis leads to many other interesting conclusions, mainly concerning the stationary and conformal measures. Moreover, several questions are left open. The methods work as well for codimensio…
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.
In this report, we first recall the Poincaré's classification theorem for minimal orientation-preserving homeomorphisms on the circle and the Ghys' classification theorem for minimal orientation-preserving group actions on the circle. Then we introduce a classification theorem for a specified class of topologically transitive orientation-preserving group actions on the circle by $\mathbb Z^d$. Also, some groups that admit/admit no topologically transitive actions on the line are determined.
It is shown that if $f$ and $g$ are any two analytic critical circle mappings with the same irrational rotation number, then the conjugacy that maps the critical point of $f$ to that of $g$ has regularity $C^{1+α}$ at the critical point, with a universal value of $α>0$. As a consequence, a new proof of the hyperbolicity of the full renormalization horseshoe of critical circle maps is given.
We will give an example of a smooth free action of $S^1=U(1)$ on $S^7$ whose orbits have unbounded lenghts (equivalently: unbounded periods). As an application of this example we construct a $C^{\infty}$ vector field $X$, defined in a neighbourhood $U$ of $0 \in \mathbb{R}^8$, such that: $U-\{0\}$ is foliated by closed integral curves of $X$, the differential $DX(0)$ at $0$ defines a $1$-parameter group of nondegenerate rotations and $X$ is {\it not} orbitally equivalent to its linearization. This proves in the $C^{\infty}$ category that the classical Poincaré Centre Theorem, true for planar n…
We investigate the structure of circle actions with the Rokhlin property, particularly in relation to equivariant $KK$-theory. Our main results are $\mathbb{T}$-equivariant versions of celebrated results of Kirchberg: any Rokhlin action on a separable, nuclear C*-algebra is $KK^\mathbb{T}$-equivalent to a Rokhlin action on a Kirchberg algebra; any Rokhlin action on an exact separable C*-algebra embeds equivariantly into $\mathcal{O}_2$ (with its unique Rokhlin action); and two circle actions with the Rokhlin property on a Kirchberg algebra are conjugate if and only if they are $KK^\mathbb{T}$-…
A nonrelativistic particle on a circle and subject to a cos^{-2}(k phi) potential is related to the two-dimensional (dihedral) Coxeter system I_2(k), for k in N. For such `dihedral systems' we construct the action-angle variables and establish a local equivalence with a free particle on the circle. We perform the quantization of these systems in the action-angle variables and discuss the supersymmetric extension of this procedure. By allowing radial motion one obtains related two-dimensional systems, including A_2, BC_2 and G_2 three-particle rational Calogero models on R, which we also analyz…
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…
Suppose G is a connected, simple, real Lie group with real rank at least two, M is an ergodic G-space with invariant probability measure, and f is a Homeo(T)-valued Borel cocycle, where Homeo(T) denotes the group of homeomorphisms of the circle T. We use an argument of E.Ghys to show that there is a G-invariant probability measure on the skew product of M and T. Furthermore, if the image of f consists of diffeomorphisms, then there is an invariant measure that is equivalent to the product measure; therefore, f is cohomologous to a cocycle with values in the isometry group of T.
