In this work we study the renormalization operator acting on piecewise smooth homeomorphisms on the circle, that turns out to be essentially the study of Rauzy-Veech renormalizations of generalized interval exchanges maps with genus one. In particular we show that renormalizations of such maps with zero mean nonlinearity and satisfying certain smoothness and combinatorial assumptions converges to the set of piecewise affine interval exchange maps.
Brin and Squier described how to classify the elements of the group of piecewise linear homeomorphisms of the real line according to conjugacy. We supply a short account of the key step in their classification. The results in this document are unoriginal. The proofs are a little sketchy in places.
We show that the stationary measure for some random systems of two piecewise affine homeomorphisms of the interval is singular, verifying partially a conjecture by Alsedà and Misiurewicz and contributing to a question of Navas on the absolute continuity of stationary measures, considered in the setup of semigroups of piecewise affine circle homeomorphisms. We focus on the case of resonant boundary derivatives.
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 study invariant ergodic measures for quasiperiodically forced circle homeomorphisms and derive that either the system is uniquely ergodic or any such measure is associated to some invariant multigraph.
Let $G$ be the group ${\rm PAff}_+({\bf S}^1)$ of piecewise--affine circle homeomorphisms or the group ${\Diff}^{\infty}(\mathbb R/\mathbb Z)$ of smooth circle diffeomorphisms. A constructive proof that all irrational rotations are distorted in $G$ is given.
We find conditions for two piecewise C^{2+ν} homeomorphisms f and g of the circle to be C^1 conjugate. Besides the restrictions on the combinatorics of the maps (we assume that the maps have bounded combinatorics), and necessary conditions on the one-side derivatives of points where f and g are not differentiable, we also assume zero mean nonlinearity for f and g.
We provide a classification of random orientation-preserving homeomorphisms of $\mathbb{S}^1$, up to topological conjugacy of the random dynamical systems generated by i.i.d. iterates of the random homeomorphism. This classification covers all random circle homeomorphisms for which the noise space is a connected Polish space and an additional extremely weak condition is satisfied.
In this paper we will refine Sacksteder's theorem for groups of orientation-preserving homeomorphisms of the circle in the case that there exists a finite orbit set. We will give a categorization of the topological possibilities for the orbits of points of the circle, along with examples.
We study renormalizations of piecewise smooth homeomorphisms on the circle, by considering such maps as generalized interval exchange maps of genus one. Suppose that $Df$ is absolutely continuous on each interval of continuity and $D\ln{Df}\in \mathbb{L}_{p}$ for some $p>1$. We prove, that under certain combinatorial assumptions on $f_{1}$ and $f_{2}$, corresponding renormalizations approach to each other in $C^{1+L_{1}}$-norm.
We investigate the curves in the complex plane which are generated by sequences of real numbers being the lifts of the points on the orbit of an orientation preserving circle homeomorphism. Geometrical properties of these curves such as boundedness, superficiality, local discrete radius of curvature are linked with dynamical properties of the circle homeomorphism which generates them: rotation number and its continued fraction expansion, existence of a continuous solution of the corresponding cohomological equation and displacement sequence along the orbit.
We study the dynamics of piecewise affine surface homeomorphisms from the point of view of their entropy. Under the assumption of positive topological entropy, we establish the existence of finitely many ergodic and invariant probability measures maximizing entropy and prove a multiplicative lower bound for the number of periodic points. This is intended as a step towards the understanding of surface diffeomorphisms. We proceed by building a jump transformation, using not first returns but carefully selected "good" returns to dispense with Markov partitions. We control these good returns throu…
We prove that the polynomial entropy of an orientation preserving homeomorphism of the circle equals 1 when the homeomorphism is not conjugate to a rotation and that it is 0 otherwise. In a second part we prove that the polynomial entropy of a flow on the two dimensional torus associated with a $C^1$ nonvanishing vector field is less that $1$. We moreover prove that when the flow possesses periodic orbits its polynomial entropy equals 1 unless it is conjugate to a rotation (in this last case, the polynomial entropy is zero).
The rigidity theory for circle homeomophisms with breaks was studied intensively in the last 20 years. It was proved that under mild conditions of the Diophantine type on the rotation number any two $C^{2+α}$ smooth circle homeomorphisms with a break point are $C^1$ smoothly conjugate to each other, provided that they have the same rotation number and the same size of the break. In this paper we prove that the conjugacy may not be $C^{2-ν}$ even if the maps are analytic outside of the break points. This result shows that the rigidity theory for maps with singularities is very different from th…
Let $f$ be an orientation preserving homeomorphisms on the circle with several break points, that is, its derivative $Df$ has jump discontinuities at these points. We study Rauzy-Veech renormalizations of piecewise smooth circle homeomorphisms, by considering such maps as generalized interval exchange maps with genus one. Suppose that $Df$ is absolutely continuous on the each interval of continuity and $D\ln{Df}\in \mathbb{L}_{p}$ for some $p>1$. We prove that, under certain combinatorial assumptions on $f$, renormalizations $R^{n}(f)$ are approximated by piecewise Möbus functions in $C^{1+L_{…
In this work we study the following realization problem: given a piecewise homeomorphism $Φ: S^1 \rightarrow S^1$, which geometrical and dynamical conditions on $Φ$ are sufficient to realize it as a Bowen-Series-like map associated to a surface group, seen as a discrete subgroup of ${\rm Homeo}^+ (S^1)$?
We describe the relation between two characterizations of conjugacy in groups of piecewise-linear homeomorphisms, discovered by Brin and Squier in [2] and Kassabov and Matucci in [5]. Thanks to the interplay between the techniques, we produce a simplified point of view of conjugacy that allows us to easily recover centralizers and lends itself to generalization.
We study the coarse geometry of the moduli space of dilation tori with two singularities and the dynamical properties of the action of the Teichmuller flow on this moduli space. This leads to a proof that the vertical foliation of a dilation torus is almost always Morse-Smale. As a corollary, we get that the generic piecewise affine circle homeomorphism with two break points -with respect to the Lebesgue measure- is Morse-Smale.
We give a complete description of the behaviour of the sequence of displacements $η_n(z)=Φ^n(x) - Φ^{n-1}(x) \ \rmod \ 1$, $z=\exp(2π\rmi x)$, along a trajectory $\{φ^{n}(z)\}$, where $φ$ is an orientation preserving circle homeomorphism and $Φ:\mathbb{R} \to \mathbb{R}$ its lift. If the rotation number $\varrho(φ)=\frac{p}{q}$ is rational then $η_n(z)$ is asymptotically periodic with semi-period $q$. This convergence to a periodic sequence is uniform in $z$ if we admit that some points are iterated backward instead of taking only forward iterations for all $z$. If $\varrho(φ) \notin \mathbb{Q…
Poincare's classification of the dynamics of homeomorphisms of the circle is one of the earliest, but still one of the most elegant, classification results in dynamical systems. Here we generalize this to quasiperiodically forced circle homeomorphisms, which have been the subject of considerable interest in recent years. Herman already showed two decades ago that a unique rotation number exists for all orbits in the quasiperiodically forced case. However, unlike the unforced case, no a priori bounds exist for the deviations from the average rotation. This plays an important role in the attempt…
The group of piecewise projective homeomorphisms of the line provides straightforward counter-examples to the so-called von Neumann conjecture. The examples are so simple that many additional properties can be established.
For a certain class of groups of piecewise linear homeomorphisms of the interval, we prove that they admit no sufficiently regular faithful action on the line. Building on previous work of Brum, Matte Bon, Rivas, and the author [arXiv:2104.14678], the new ingredient is an observation from a recent work of Hyde and Tatch Moore [arXiv:2103.14911], which allows to reduce the problem to the case of the circle, and then apply Herman-Yoccoz theory.
Let f be a class P -homeomorphism of the circle. We prove that there exists a piecewise analytic homeomorphism that conjugate f to a one-class P with prescribed break points lying on pairwise distinct orbits. As a consequence, we give a sharp estimate for the smoothness of a conjugation of class P -homeomorphism f of the circle satisfying the (D)-property (i.e. the product of f-jumps in the break points contained in a same orbit is trivial), to diffeomorphism. When f does not satisfy the (D)-property the conjugating homeomorphism is never piecewise C^1 and even more it is not absolutely contin…
An element of a group is \emph{reversible} if it is conjugate to its own inverse, and it is \emph{strongly reversible} if it is conjugate to its inverse by an involution. A group element is strongly reversible if and only if it can be expressed as a composite of two involutions. In this paper the reversible maps, the strongly reversible maps, and those maps that can be expressed as a composite of involutions are determined in certain groups of piecewise linear homeomorphisms of the real line.
Let $f_i\in C^{2+α}(S^1\setminus \{a_i,b_i\}), α>0, i=1,2$ be circle homeomorphisms with two break points $a_i,b_i$, i.e. discontinuities in the derivative $f_i$, with identical irrational rotation number $rho$ and $μ_1([a_1,b_1])= μ_2([a_2,b_2])$, where $μ_i$ are invariant measures of $f_i$. Suppose the products of the jump ratios of $Df_1$ and $Df_2$ do not coincide, i.e. $\frac{Df_1(a_1-0)}{Df_1(a_1+0)}\times \frac{Df_1(b_1-0)}{Df_1(b_1+0)}\neq \frac{Df_2(a_2-0)}{Df_2(a_2+0)}\times \frac{Df_2(b_2-0)}{Df_2(b_2+0)}$. Then the map $ψ$ conjugating $f_1$ and $f_2$ is a singular function, i.e. it…
As was known to H. Poincare, an orientation preserving circle homeomorphism without periodic points is either minimal or has no dense orbits, and every orbit accumulates on the unique minimal set. In the first case the minimal set is the circle, in the latter case a Cantor set. In this paper we study a two-dimensional analogue of this classical result: we classify the minimal sets of non-resonant torus homeomorphisms; that is, torus homeomorphisms isotopic to the identity for which the rotation set is a point with rationally independent irrational coordinates.
We study random independent and identically distributed iterations of functions from an iterated function system of homeomorphisms on the circle which is minimal. We show how such systems can be analyzed in terms of iterated function systems with probabilities which are non-expansive on average.
We show that a generic quasiperiodically forced circle homeomorphism is mode-locked: the rotation number in the fibres is rationally related to the rotation number in the base and it is stable under small perturbations of the system. As a consequence, this implies that for a generic parameter family of quasiperiodically forced circle homeomorphisms satisfying a twist condition, the graph of the rotation number as a function of the parameter is a devil's staircase.
We show that given a compact minimal system $(X,g)$ and an element $h$ of the topological full group $τ[g]$ of $g$, then the infinite orbits of $h$ admit a locally constant orientation with respect to the orbits of $g$. We use this to obtain a clopen partition of $(X,G)$ into minimal and periodic parts, where $G$ is any virtually polycyclic subgroup of $τ[g]$. We also use the orientation of orbits to give a refinement of the index map and to describe the role in $τ[g]$ of the submonoid generated by the induced transformations of $g$. Finally, we consider the problem, given a homeomorphism $h$ …
Classical results by Poincaré and Denjoy show that two orientation-preserving $C^2$ diffeomorphisms of the circle are topologically conjugate if and only if they have the same rotation number. We show that there is no possibility of getting such a complete numerical Borel invariant for the conjugacy relation of orientation-preserving circle diffeomorphisms by homeomorphisms with higher degree of regularity. For instance, we consider conjugacy by Hölder homeomorphisms or by $C^k$-diffeomorphisms with $k\in \mathbb{Z}^+ \cup \{\infty\}$. The proof combines techniques from Descriptive Set Theory …
It is known that piecewise affine surface homeomorphisms always have measures of maximal entropy. This is easily seen to fail in the discontinuous case. Here we describe a piecewise affine, globally continuous surface map with no measure of maximal entropy.
In this article we study the existence of limit cycles in families of piecewise smooth differential equations having the unit circle as discontinuity region. We consider families presenting singularities of center or saddle type, visible or invisible, as well as the case without singularities. We establish an upper bound for the number of limit cycles and give examples showing that the maximum number of limit cycles can be reached. We also discuss the existence of homoclinic cycles for such differential equations in the saddle-center case.
Let $T_{f}$ be a circle homeomorphism with two break points $a_{b},c_{b}$ and irrational rotation number $\varrho_{f}$. Suppose that the derivative $Df$ of its lift $f$ is absolutely continuous on every connected interval of the set $S^{1}\backslash\{a_{b},c_{b}\}$, that $DlogDf \in L^{1}$ and the product of the jump ratios of $ Df $ at the break points is nontrivial, i.e. $\frac{Df_{-}(a_{b})}{Df_{+}(a_{b})}\frac{Df_{-}(c_{b})}{Df_{+}(c_{b})}\neq1$. We prove that the unique $T_{f}$- invariant probability measure $μ_{f}$ is then singular with respect to Lebesgue measure $l$ on $S^{1}$.
We develop a renormalization theory for analytic homeomorphisms of the circle with two cubic critical points. We prove a renormalization hyperbolicity theorem. As a basis for the proofs, we develop complex a priori bounds for multi-critical circle maps.
Let $f$ and $g$ be two class $P$-homeomorphisms of the circle $S^{1}$ with break points singularities. Assume that the derivatives $\textrm{Df}$ and $\textrm{Dg}$ are absolutely continuous on every continuity interval of $\textrm{Df}$ and $\textrm{Dg}$ respectively. Denote by $C(f)$ the set of break points of $f$. For $c\in S^{1}$, denote by $π_{s, O_{f}(c)}(f)$ the product of $f-$ jumps in break points lying to the $f-$ orbit of $c$ and by $\textrm{SO}(f) = \{O_{f}(c):~c \in C(f)~\textrm{and}~π_{s, O_{f}(c)}(f)\neq 1\}$, called the set of singular $f$-orbits. The maps $f$ and $g$ are called b…
Let $f_{i},$ $i=1,2$ be piecewise-smooth $C^{1}$ circle homeomorphisms with two break points, $\log Df_{i},$ $i=1,2$ are absolutely continuous on each continuity intervals of $Df_{i}$ and $D\log Df_{i}\in L^{p}$ for some $p>1.$ Suppose, the jump ratios of $f_{1} $ and $f_{2} $ at their break points do not coincide but have the same total jumps (i.e. the product of jump ratios) and identical irrational rotation number of bounded type. Then the conjugation $h$ between $f_{1} $ and $f_{2} $ is a singular function, i.e. it is continuous on $S^1,$ but $Dh(x)=0$ a.e. with respect to Lebesgue measure…
We present a one-parameter family of continuous, piecewise affine, area preserving maps of the square, which are inspired by a dynamical system in game theory. Interested in the coexistence of stochastic and (quasi-)periodic behaviour, we investigate invariant annuli separated by invariant circles. For certain parameter values, we explicitly construct invariant circles both of rational and irrational rotation numbers, and present numerical experiments of the dynamics on the annuli bounded by these circles.
It is well known that the rotation number of a circle homeomorphism defined by H. Poincaré allows to completely understand the dynamics of such a map from the topological point of view. In this paper, we collect some results concerning the existence of topological dynamical models associated to rotation sets of homeomorphisms of $\T^2$.
Critical circle homeomorphisms have an invariant measure totally singular with respect to the Lebesgue measure. We prove that singularities of the invariant measure are of Holder type. The Hausdorff dimension of the invariant measure is less than 1 but greater than 0.
To every dynamical system $(X,φ)$ over a totally disconnected compact space, we associate a left-orderable group $T(φ)$. It is defined as a group of homeomorphisms of the suspension of $(X,φ)$ which preserve every orbit of the suspension flow and act by dyadic piecewise linear homeomorphisms in the flow direction. We show that if the system is minimal, the group is simple, and if it is a subshift then the group is finitely generated. The proofs of these two statements are short and elementary, providing straightforward examples of finitely generated simple left-orderable groups. We show that i…
In this paper we present a generalization of Poincaré's Rotation Theory of homeomorphisms of the circle to the case of one-dimensional compact abelian groups which are solenoidal groups, {\it i.e.}, groups which fiber over the circle with fiber a Cantor abelian group. We define rotation elements, \emph{à la} Poincaré and discuss the dynamical properties of translations on these solenoidal groups. We also study the semiconjugation problem when the rotation element generates a dense subgroup of the solenoidal group. Finally, we comment on the relation between Rotation Theory and entropy for thes…
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.
Let $f_{i},i=1,2$ be orientation preserving circle homeomorphisms with a finite number of break points, at which the first derivatives $Df_{i}$ have jumps, and with identical irrational rotation number $ρ=ρ_{f_{1}}=ρ_{f_{2}}.$ The jump ratio of $f_{i}$ at the break point $b$ is denoted by $σ_{f_{i}}(b)$, i.e. $σ_{f_{i}}(b):=\frac{Df_{i}(b-0)}{Df_{i}(b+0)}$. Denote by $σ_{f_{i}}, i=1,2,$ the total jump ratio given by the product over all break points $b$ of the jump ratios $σ_{f_{i}}(b)$ of $f_{i}$. We prove, that for circle homeomorphisms $f_{i}, i=1,2$, which are $C^{2+\varepsilon}, \varepsil…
