The present article is an extended version of our previous work posted in arXiv:1408.1663v1, entitled "Piecewise Contractions Defined by Iterated Function Systems"
Let $-1<λ<1$ and $f:[0,1)\to\mathbb{R}$ be a piecewise $λ$-affine map, that is, there exist points $0=c_0<c_1<\cdots <c_{n-1}<c_n=1$ and real numbers $b_1,\ldots,b_n$ such that $f(x)=λx+b_i$ for every $x\in [c_{i-1},c_i)$. We prove that, for Lebesgue almost every $δ\in\mathbb{R}$, the map $f_δ=f+δ\,({\rm mod}\,1)$ is asymptotically periodic. More precisely, $f_δ$ has at most $2n$ periodic orbits and the $ω$-limit set of every $x\in [0,1)$ is a periodic orbit.
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.
A class of piecewise affine hyperbolic maps on a bounded subset of the plane is considered. It is shown that if a map from this class is sufficiently area-expanding then almost surely this map has an absolutely continuous invariant measure.
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.
We prove that for continuous maps on the interval, the existence of an n-cycle, implies the existence of n-1 points which interwind the original ones and are permuted by the map. We then use this combinatorial result to show that piecewise affine maps (with no zero slope) cannot be infinitely renormalizable.
We use canonical Markov extensions (Hofbauer towers) to give an explicit construction of the natural extensions of various measure preserving endomorphisms, and present some applications to particular examples.
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.
Consider a multimodal interval map $f$ of $C^3$ with non-flat critical points. We establish several characterizations of the map $f$ is quasi-symmetrically conjugated to a piecewise affine map in the case $f$ is topologically exact and all of its periodic points are hyperbolic repelling. In particular, we give a negative answer to a question posted by Henk Bruin in \cite{Bru96}.
We construct a chaotic discrete-time continuous-state Hopfield network with piecewise-affine nonnegative activation functions and weight matrix with small positive entries. More precisely, there exists a Cantor set $C$ in the state space such that the network has sensitive dependence on initial conditions at initial states in $C$ and the network orbit of each initial state in $C$ has $C$ as its $ω$-limit set. The approach we use is based on tools developed and employed recently in the study of the topological dynamics of piecewise-contractions. The parameters of the chaotic network are explici…
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 consider a class of simple one parameter families of interval maps, and we study how metric (resp. topological) entropy changes as the parameter varies. We show that in many cases the entropy displays a semi-regular behaviour, i.e. it is smooth on an open and dense set. This feature is due to a combinatorial property called matching
In this paper, we study the dimension theory of a class of piecewise affine systems in euclidean spaces suggested by Michael Barnsley, with some applications to the fractal image compression. It is a more general version of the class considered in the work of Keane, Simon and Solomyak [The dimension of graph directed attractors with overlaps on the line, with an application to a problem in fractal image recognition. {\it Fund. Math.}, {\bf 180}(3):279-292, 2003] and can be considered as the continuation of the works [On the dimension of self-affine sets and measures with overlaps. {\it Proc. A…
We consider the growth of heights of the points of the orbits of (piecewise) affine maps of the plane, with rational parameters. We analyse the asymptotic growth rate of both global and local ($p$-adic) heights, for the primes $p$ that divide the parameters. We show that almost all the points in a domain of linearity (such as an elliptic island in an area-preserving map) have the same exponential growth rate. We also show that the convergence of the $p$-adic height may be non-uniform, with arbitrarily large fluctuations occurring arbitrarily close to any point. We explore numerically the behav…
Let $I=[0,1)$, $-1<λ<1$ and $f\colon I\to I$ be a piecewise $λ$-affine map of the interval $I$, i.e., there exist a partition $0=a_0<a_1<\cdots< a_{k-1}<a_k=1$ of the interval $I$ into $k\geq2$ subintervals and $b_1,\ldots, b_k\in\mathbb{R}$ such that $f(x)=λx+ b_i$ for every $x\in[a_{i-1},a_{i})$ and $i=1,\ldots,k$. The exceptional set $\mathcal{E}_f$ of $f$ is the set of parameters $δ\in\mathbb{R}$ such that $R_δ\circ f$ is not asymptotically periodic, where $R_δ\colon I\to I$ is the rotation of angle $δ$. In this paper we prove that $\mathcal{E}_f$ has zero Hausdorff dimension. We derive th…
For flows, the singular cycles connecting saddle periodic orbit and saddle equilibrium can poten- tially result in the so-called singular horseshoe, which means the existence of a non-uniformly hyperbolic chaotic invariant set. However, it is very hard to find a specific dynamical system that exhibits such singular cycles in general. In this paper, the existence of the singular cycles involved in saddle periodic orbits is studied by two types of piecewise affine systems: one is the piecewise affine system having an admissible saddle point with only real eigenvalues and an admissible saddle per…
We study maps of the unit interval whose graph is made up of two increasing segments and which are injective in an extended sense. Such maps $f_{\p}$ are parametrized by a quintuple $\p$ of real numbers satisfying inequations. Viewing $f_{\p}$ as a circle map, we show that it has a rotation number $ρ(f_{\p})$ and we compute $ρ(f_{\p})$ as a function of $\p$ in terms of Hecke-Mahler series. As a corollary, we prove that $ρ(f_{\p})$ is a rational number when the components of $\p$ are algebraic numbers.
Mixed-mode oscillations (MMOs) are complex oscillatory patterns in which large-amplitude relaxation oscillations (LAOs) alternate with small-amplitude oscillations (SAOs). MMOs are found in singularly perturbed systems of ordinary differential equations of slow-fast type, and are typically related to the presence of so-called folded singularities and the corresponding canard trajectories in such systems. Here, we introduce a canonical family of three-dimensional slow-fast systems that exhibit MMOs which are induced by relaxation-type dynamics, and which are hence based on a "jump mechanism", r…
The topological entropy of piecewise affine maps is studied. It is shown that singularities may contribute to the entropy only if there is angular expansion and we bound the entropy via the expansion rates of the map. As a corollary we deduce that non-expanding conformal piecewise affine maps have zero topological entropy. We estimate the entropy of piecewise affine skew-products. Examples of abnormal entropy growth are provided.
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.
We present sufficient conditions for the (strong) statistical stability of some classes of multidimensional piecewise expanding maps. As a consequence we get that a certain natural two-dimensional extension of the classical one-dimensional family of tent maps is statistically stable.
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.
Let $g:\, [0, 1]\rightarrow [0, 1]$ be piecewise linear unimodal map. We say that $g$ has non-trivial piecewise linear commutator, if there is a continuous piecewise linear $ψ:\, [0, 1]\rightarrow [0, 1]$ such that $g\circ ψ= ψ\circ g$, and, moreover, $ψ$ is neither an iteration of $g$, not a constant map. We prove that if $g$ has a non-trivial piecewise linear commutator, then $g$ is topologically conjugated with the tent map by a piecewise linear conjugacy.
We construct inducing schemes for general multi-dimensional piecewise expanding maps where the base transformation is Gibbs-Markov and the return times have exponential tails. Such structures are a crucial tool in proving statistical properties of dynamical systems with some hyperbolicity.
This paper gives the definition of the Conley index for a piecewise continuous map, which is only well defined on compatible isolating neighborhoods with Wazewski property slightly weaker than continuous situation.
For any integer $r$ with $1\leq r<\infty$, we present a one-parameter family $F_σ$ $(0<σ<1)$ of 2-dimensional piecewise $\mathcal C^r$ expanding maps such that each $F_σ$ has an observable (i.e. Lebesgue positive) Lyapunov irregular set. These maps are obtained by modifying the piecewise expanding map given in Tsujii (2000). In strong contrast to it, we also show that any Lyapunov irregular set of any 2-dimensional piecewise real analytic expanding map is not observable. This is based on the spectral analysis of piecewise expanding maps in Buzzi (2000).
We study the complexity of the itineraries of injective piecewise contracting maps on the interval. We prove that for any such map the complexity function of any itinerary is eventually affine. We also prove that the growth rate of the complexity is bounded from above by the number $N-1$ of discontinuities of the map. To show that this bound is optimal, we construct piecewise affine contracting maps whose itineraries all have the complexity $(N-1)n +1$. In these examples, the asymptotic dynamics takes place in a minimal Cantor set containing all the discontinuities.
We prove a fiberwise almost sure invariance principle for random piecewise expanding transformations in one and higher dimensions using recent developments on martingale techniques.
Let $f : [0,1)\rightarrow [0,1)$ be a $2$-interval piecewise affine increasing map which is injective but not surjective. Such a map $f$ has a rotation number and can be parametrized by three real numbers. We make fully explicit the dynamics of $f$ thanks to two specific functions $δ$ and $ϕ$ depending on these parameters whose definitions involve Hecke-Mahler series. As an application, we show that the rotation number of $f$ is rational, when the three parameters are algebraic numbers.
We construct an appropriate metric on the collection of piecewise $\mathcal C^r$ maps defined on a compact interval. Although this metric space turns out to be not complete, we show that it is indeed a Baire space. As an application, we prove the robustness of non-degenerate critical points for this class of systems and we show the genericity of piecewise Lipschitz maps that admit an invariant Borel probability measure.
For each piecewise linear Lorenz map that expand on average, we show that it admits a dichotomy: it is either periodic renormalizable or prime. As a result, such a map is conjugate to a $β$-transformation.
In this paper, we introduce a generalized piecewise translation map on the Euclidean space. We provide a special case when this map is always of finite type. For a finite type map in this case, we form conjectures on the semi-continuity of the attractor. Moreover, we also provide some conjectures and pictures on piecewise translations on a two dimensional torus.
For a two parameter family of two-dimensional piecewise linear maps and for every natural number $ n $ we prove not only the existence of intervals of parameters for which the respective maps are $ n $ times renormalizable but also we show the existence of intervals of parameters where the coexistence of at least $ 2^n $ strange attractors takes place. This family of maps contains the two-dimensional extension of the classical one-dimensional family of tent maps.
We introduce a weak transversality condition for piecewise C^{1+α} and piecewise hyperbolic maps which admit a C^{1+α} stable distribution. We show good bounds on the essential spectral radius of the associated transfer operators acting on classical anisotropic Sobolev spaces of Triebel-Lizorkin type. In many cases, we obtain a spectral gap from which we deduce the existence of finitely many physical measures with basin of total measure. The analysis relies on standard techniques (in particular complex interpolation) and applies also to piecewise expanding maps and to Anosov diffeomorphisms, g…
We prove that the Newton quotient of the average R(t) of a lipschitzian function (with non vanishing variation) with respect to the SRB measure on a transversal family f_t of piecewise expanding unimodal maps, after an appropriated normalization, converges in distribution to the normal distribution.
We study some special classes of piecewise continuous maps on a finite smooth partition of a compact manifold and look for invariant measures for such maps. We show that in the simplest one-dimensional case (so-called interval translation maps) a Borel probability non-atomic invariant measure exists for any map. We use this result to demonstrate that any interval translation map endowed with such a measure is metrically equivalent to an interval exchange map. Finally, we study the general case of piecewise continuous maps and prove a simple result on existence of an invariant measure provided …
We consider piecewise $C^2$ non-flat maps of the interval and show that, for Lebesgue almost every point, its omega-limit set is either a periodic orbit, a cycle of intervals or the closure of the orbits of a subset of the critical points. In particular, every piecewise $C^2$ non-flat map of the interval displays only a finite number of non-periodic attractors.
We consider a class of piecewise hyperbolic maps from the unit square to itself preserving a contracting foliation and inducing a piecewise expanding quotient map, with infinite derivative (like the first return maps of Lorenz like flows). We show how the physical measure of those systems can be rigorously approximated with an explicitly given bound on the error, with respect to the Wasserstein distance. We apply this to the rigorous computation of the dimension of the measure. We present a rigorous implementation of the algorithms using interval arithmetics, and the result of the computation …
We determine periodic and aperiodic points of certain piecewise affine maps in the Euclidean plane. Using these maps, we prove for $λ\in\{\frac{\pm1\pm\sqrt5}2,\pm\sqrt2,\pm\sqrt3\}$ that all integer sequences $(a_k)_{k\in\mathbb Z}$ satisfying $0\le a_{k-1}+λa_k+a_{k+1}<1$ are periodic.
We investigate the properties of absolutely continuous invariant probability measures (ACIPs), especially those measures with bounded variation densities, for piecewise area preserving maps (PAPs) on $\mathbb{R}^d$. This class of maps unifies piecewise isometries (PWIs) and piecewise hyperbolic maps where Lebesgue measure is locally preserved. Using a functional analytic approach, we first explore the relationship between topological transitivity and uniqueness of ACIPs, and then give an approach to construct invariant measures with bounded variation densities for PWIs. Our results "partially"…
We study periodic, piecewise linear maps on the plane starting with the Mort Brown's map. We show that if the number of pieces is two, there is only a short list of possible periods (this fact can be seen as the crystallographic restriction for this class of maps). Otherwise, without the restriction on the number of pieces, a map can have any period. We show how to construct such maps using binary trees and so called admissible sequences.
We study a certain class of piecewise monotonic maps of interval. These maps are strictly monotone on finite interval partition, satisfies Markov condition and have generator property. We show that for a function from this class distributional chaos is always present and we study its basic properties. Main result states that distributional spectrum as well as weak spectrum is always finite. This is a generalization of same result for continuous maps on the interval, circle and tree. Examples showing that conditions on mentions class can not be weakened are presented.
In this paper, we introduce a family of second-order sigma delta quantization schemes for analog-to-digital conversion which are `quiet' : quantization output is guaranteed to fall to zero at the onset of vanishing input. In the process, we prove that the origin is a globally attractive fixed point for the related family of asymmetrically-damped piecewise affine maps. Our proof of convergence is twofold: first, we construct a trapping set using a Lyapunov-type argument; we then take advantage of the asymmetric structure of the maps under consideration to prove convergence to the origin from wi…
The stable and unstable manifolds of an invariant set of a piecewise-smooth map are themselves piecewise-smooth. Consequently, as parameters of a piecewise-smooth map are varied, an invariant set can develop a homoclinic connection when its stable manifold intersects a non-differentiable point of its unstable manifold (or vice-versa). This is a codimension-one bifurcation analogous to a homoclinic tangency of a smooth map, referred to here as a homoclinic corner. This paper presents an unfolding of generic homoclinic corners for saddle fixed points of planar piecewise-smooth continuous maps. I…
We show the existence of Lebesgue-equivalent conservative and ergodic $σ$-finite invariant measures for a wide class of one-dimensional random maps consisting of piecewise convex maps. We also estimate the size of invariant measures around a small neighborhood of a fixed point where the invariant density functions may diverge. Application covers random intermittent maps with critical points or flat points. We also illustrate that the size of invariant measures tends to infinite for random maps whose right branches exhibit a strongly contracting property on average, so that they have a strong r…
One of the most basic, longstanding open problems in the theory of dynamical systems is whether reachability is decidable for one-dimensional piecewise affine maps with two intervals. In this paper we prove that for injective maps, it is decidable. We also study various related problems, in each case either establishing decidability, or showing that they are closely connected to Diophantine properties of certain transcendental numbers, analogous to the positivity problem for linear recurrence sequences. Lastly, we consider topological properties of orbits of one-dimensional piecewise affine ma…
For any transitive piecewise monotonic map for which the set of periodic measures is dense in the set of ergodic invariant measures (such as monotonic mod one transformations and piecewise monotonic maps with two monotonic pieces), we show that the set of points for which the Birkhoff average of a continuous function does not exist (called the irregular set) is either empty or has full topological entropy. This generalizes Thompson's theorem for irregular sets of $β$-transformations, and reduces a complete description of irregular sets of transitive piecewise monotonic maps to Hofbauer-Raith p…
We consider some classes of piecewise expanding maps in finite dimensional spaces having invariant probability measures which are absolutely continuous with respect to Lebesgue measure. We derive an entropy formula for such measures and, using this entropy formula, in some parametrized families we present sufficient conditions for the continuity of that entropy with respect to the parameter. We apply our results to a classical one-dimensional family of tent maps and a family of two-dimensional maps which arises as the limit of return maps when a homoclinic tangency is unfolded by a family of t…
We consider piecewise expanding maps of the interval with finitely many branches of monotonicity and show that they are generically combinatorially stable, i.e., the number of ergodic attractors and their corresponding mixing periods do not change under small perturbations of the map. Our methods provide a topological description of the attractor and, in particular, give an elementary proof of the density of periodic orbits.
In this paper we consider piecewise affine differential equations modeling gene networks. We work with arbitrary decay rates, and under a local hypothesis expressed as an alignment condition of successive focal points. The interaction graph of the system may be rather complex (multiple intricate loops of any sign, multiple thresholds...). Our main result is an alternative theorem showing that, if a sequence of region is periodically visited by trajectories, then under our hypotheses, there exists either a unique stable periodic solution, or the origin attracts all trajectories in this sequence…
In two-parameter bifurcation diagrams of piecewise-linear continuous maps on $\mathbb{R}^N$, mode-locking regions typically have points of zero width known as shrinking points. Near any shrinking point, but outside the associated mode-locking region, a significant proportion of parameter space can be usefully partitioned into a two-dimensional array of nearly-hyperbolic annular sectors. The purpose of this paper is to show that in these sectors the dynamics is well-approximated by a three-parameter family of skew sawtooth circle maps, where the relationship between the skew sawtooth maps and t…
Mode-locking regions (resonance tongues) formed by border-collision bifurcations of piecewise-smooth, continuous maps commonly exhibit a distinctive sausage-like geometry with pinch points called "shrinking points". In this paper we extend our unfolding of the piecewise-linear case [{\em Nonlinearity}, 22(5):1123-1144, 2009] to show how shrinking points are destroyed by nonlinearity. We obtain a codimension-three unfolding of this shrinking point bifurcation for $N$-dimensional maps. We show that the destruction of the shrinking points generically occurs by the creation of a curve of saddle-no…
We consider a specific piecewise rotation of the plane that is continuous on two half-planes, as studied by some authors like Boshernitzan, Goetz and Quas. If the angle belongs to the set $\{\fracπ{2},\frac{2π}{3},\fracπ{4}\}$, we give a complete description of the symbolic dynamics of this map in the non symmetric bijective case.
We devote this work to the functional equation $ψ\circ g = g\circ ψ$, where $ψ$ is an unknown function and $g$ is piecewise linear unimodal map, which is topologically conjugated to the tent map. We will call such $ψ$ self-semiconjugations of $g$. Our the main results are the following: 1. Suppose that there is a self-semiconjugation of $g$, whose tangent at $0$ is not a power of $2$, and suppose that all the kinks of $g$ are in the complete pre-image of $0$. Then all the self-semiconjugations of $g$ are piecewise linear. 2. Suppose that all self-semiconjugations of $g$ are piecewise linear. T…
