Property FW is a natural combinatorial weakening of Kazhdan's Property T. We prove that the group of piecewise homographic self-transformations of the real projective line, has "few" infinite subgroups with Property FW. In particular, no such subgroup is amenable or has Kazhdan's Property T. These results are extracted from a longer paper. We provide a complete proof, whose main tools are the use of the notion of partial action and of one-dimensional geometric structures.
We give a condition on a piecewise constant roof function and an irrational rotation by $α$ on the circle to give rise to a special flow having the mild mixing property. Such flows will also satisfy Ratner's property. As a consequence we obtain a class of mildly mixing singular flows on the two-torus that arise from quasi-periodic Hamiltonians flows by velocity changes.
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.
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.
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.
We study the near action of the group PC of piecewise continuous self-transformations of the circle. Elements of this group are only defined modulo indeterminacy on a finite subset, which raises the question of realizability: a subgroup of PC is said to be realizable if it can be lifted to a group of permutations of the circle. We show that every finitely generated abelian subgroup of PC is realizable. We show that this is not true for arbitrary subgroups, by exhibiting a non-realizable finitely generated subgroup of the group of interval exchanges with flips. The group of (oriented) interval …
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.
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…
Let $I=[0,1)$ and $\mathcal{PC}(I)$ [resp. $\mathcal{PC}^+(I)$] be the quotient group of the group of all piecewise continuous [resp. piecewise continuous and orientation preserving] bijections of $I$ by its normal subgroup consisting in elements with finite support (i.e. that are trivial except at possibly finitely many points). Unpublished Theorems of Arnoux ([Arn81b]) state that $\mathcal{PC}^+(I)$ and certain groups of interval exchanges are simple, their proofs are the purpose of the Appendix. Dealing with piecewise direct affine maps, we prove the simplicity of the group $\mathcal A^+(I)…
When high-dimensional non-uniformly hyperbolic chaotic systems undergo dynamical perturbations, their long-time statistics are generally observed to respond differentiably with respect to the perturbation. Although important in applications, this differentiability, which is thought to be connected to the dimensionality of the system, has remained resistant to rigorous study. To model non-uniformly hyperbolic systems, we consider a family of the mathematically tractable class of piecewise smooth hyperbolic maps, the Lozi maps. For these maps we prove that the existence of a formal derivative of…
We study the dynamics of piecewise conformal maps in the Riemann sphere. The normality and chaotic regions are defined and we state several results and properties of these sets. We show that the stability of these piecewise maps is related to the Kleinian group generated by their transformations under certain hypotheses. The general motivation of the article is to compare the dynamics of piecewise conformal maps and those of the Kleinian groups and iterations of rational maps.
About the Shadowing Theorem for piecewise smooth vector fields with sliding motion
Universidade Estadual Paulista UNESP（圣保罗州立大学）；Departamento de Matemática, Instituto de Biociências, Letras e Ciências Exatas, Universidade Estadual Paulista UNESP（圣保罗州立大学生物科学、文学与理学院数学系）
Let $IET(\mathbb{S}^{1})$ be the group of interval exchange transformation of $\mathbb{S}^{1}$ and $\mathcal{AC}_{+}(\mathbb{S}^{1})$ be the group of absolutely continuous preserving orientation bijection with inverse absolutely continuous. We denote by $\mathcal{ACI}$ the group generated by $IET(\mathbb{S}^{1})$ and $\mathcal{AC}_{+}(\mathbb{S}^{1})$. Given a suitable distance on $\mathcal{ACI}$, we classify all continuous homomorphisms $ρ:\mathbb{R} \to \mathcal{ACI}$. More precisely, $ρ$ is conjugated to a continuous homomorphism $\hatρ:\mathbb{R} \to \mathcal{AC}_{+}(\uplus_{i}(\mathbb{S}^…
Ghys and Sergiescu proved in the $80$s that Thompson's group $T$, and hence $F$, admits actions by $C^{\infty}$ diffeomorphisms of the circle . They proved that the standard actions of these groups are topologically conjugate to a group of $C^\infty$ diffeomorphisms. Monod defined a family of groups of piecewise projective homeomorphisms, and Lodha-Moore defined finitely presentable groups of piecewise projective homeomorphisms. These groups are of particular interest because they are nonamenable and contain no free subgroup. In contrast to the result of Ghys-Sergiescu, we prove that the group…
We construct a mixing continuous piecewise linear map on [-1,1] with the property that a two-dimensional lattice made of these maps with a linear north and east nearest neighbour coupling admits a phase transition. We also provide a modification of this construction where the local map is an expanding analytic circle map. The basic strategy is borroughed from [Gielis-MacKay (2000)], namely we compare the dynamics of the CML to those of a probabilistic cellular automaton of Toom's type.
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.
Step skew products with interval fibres and a subshift as a base are considered. It is proved that if the fibre maps are continuous, piecewise monotone, expanding and surjective and the subshift has the specification property and a periodic orbit such that the composition of the fibre maps along this orbit is mixing, then the corresponding step skew product has the specification property.
Following the works of Furstenberg and Glasner on stationary means, we strengthen and extend in this paper some recent results by Di Nasso, Goldbring, Jin, Leth, Lupini and Mahlburg on piecewise syndeticity of product sets in countable \textsc{amenable} groups to general countable measured groups. We point out several fundamental differences between the behavior of products of "large" sets in Liouville and non-Liouville measured groups. As a (very) special case of our main results, we show that if $G$ is a free group of finite rank, and $A$ and $B$ are "spherically large" subsets of $G$, then …
We study conditions under which a piecewise affine mapping has the Lipschitz shadowing property. As an application, we show that there exists a homeomorphism with a nonisolated fixed point having the Lipschitz shadowing property.
Jin proved that whenever $A$ and $B$ are sets of positive upper density in $\Z$, $A+B$ is piecewise syndetic. Jin's theorem was subsequently generalized by Jin and Keisler to a certain family of abelian groups, which in particular contains $\Z^d$. Answering a question of Jin and Keisler, we show that this result can be extended to countable amenable groups. Moreover we establish that such sumsets (or -- depending on the notation -- "productsets") are piecewise Bohr, a result which for $G=\Z$ was proved by Bergelson, Furstenberg and Weiss. In the case of an abelian group $G$, we show that a set…
We prove that if $g\geq 2$ then the set of all Abelian differentials $(M,ω)$ for which the vertical flow is mildly mixing is dense in every stratum of the moduli space $\mathcal{H}_g$. The proof is based on a sufficient condition for special flows over irrational rotations and under piecewise constant roof functions to be mildly mixing.
Any continuous, transitive, piecewise monotonic map is determined up to a binary choice by its dimension module with the associated finite sequence of generators. The dimension module by itself determines the topological entropy of any transitive piecewise monotonic map, and determines any transitive unimodal map up to conjugacy. For a transitive piecewise monotonic map which is not essentially injective, the associated dimension group is a direct sum of simple dimension groups, each with a unique state.
We study the properties of rotation numbers for some groups of piecewise linear homeomorphisms of the circle. We use these properties to obtain results on PL rigidity, non isomorphicity, non exoticity of automorphisms, non smoothability for Thompson-Stein groups.
We prove that a topologically generic network (an open and dense set of networks) of three or more inhibitory neurons have periodic behavior with a finite number of limit cycles that persist under small perturbations of the structure of the network. The network is modeled by the Poincare transformation which is piecewise continuous and locally contractive on a compact region B of a finite dimensional manifold, with the separation property: it transforms homeomorphically the different continuity pieces of B into pairwise disjoint sets.
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…
We study the global dynamics of integrate and fire neural networks composed of an arbitrary number of identical neurons interacting by inhibition and excitation. We prove that if the interactions are strong enough, then the support of the stable asymptotic dynamics consists of limit cycles. We also find sufficient conditions for the synchronization of networks containing excitatory neurons. The proofs are based on the analysis of the equivalent dynamics of a piecewise continuous Poincaré map associated to the system. We show that for strong interactions the Poincaré map is piecewise contractiv…
We prove that if a quasi-isometry of warped cones is induced by a map between the base spaces of the cones, the actions must be conjugate by this map. The converse is false in general, conjugacy of actions is not sufficient for quasi-isometry of the respective warped cones. For a general quasi-isometry of warped cones, using the asymptotically faithful covering constructed in a previous work with Jianchao Wu, we deduce that the two groups are quasi-isometric after taking Cartesian products with suitable powers of the integers. Secondly, we characterise geometric properties of a group (coarse e…
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 consider special flows over two-dimensional rotations by $(α,β)$ on $\T^2$ and under piecewise $C^2$ roof functions $f$ satisfying von Neumann's condition $\int_{\T^2}f_x(x,y)\,dx\,dy\neq 0\neq \int_{\T^2}f_y(x,y)\,dx\,dy.$ Such flows are shown to be always weakly mixing and never partially rigid. For an uncountable set of $(α,β)$ with both $α$ and $β$ of unbounded partial quotients the strong mixing property is proved to hold. It is also proved that while specifying to a subclass of roof functions and to ergodic rotations for which $α$ and $β$ are of bounded partial quotients the correspon…
We extend the notion of matching for one-dimensional dynamical systems to random matching for random dynamical systems on an interval. We prove that for a large family of piecewise affine random systems of the interval the property of random matching implies that any invariant density is piecewise constant. We further introduce a one-parameter family of random dynamical systems that produce signed binary expansions of numbers in the interval [-1,1]. This family has random matching for Lebesgue almost every parameter. We use this to prove that the frequency of the digit 0 in the associated sign…
