We consider the effect on the mixing properties of a piecewise smooth interval map $f$ when its domain is divided into $N$ equal subintervals and $f$ is composed with a permutation of these. The case of the stretch-and-fold map $f(x)=mx \bmod 1$ for integers $m \geq 2$ is examined in detail. We give a combinatorial description of those permutations $σ$ for which $σ\circ f$ is still (topologically) mixing, and show that the proportion of such permutations tends to $1$ as $N \to \infty$. We then investigate the mixing rate of $σ\circ f$ (as measured by the modulus of the second largest eigenvalue of the transfer operator). In contrast to the situation for continuous time diffusive systems, we show that composition with a permutation cannot improve the mixing rate of $f$, but typically makes it worse. Under some mild assumptions on $m$ and $N$, we obtain a precise value for the worst mixing rate as $σ$ ranges through all permutations; this can be made arbitrarily close to $1$ as $N \to \infty$ (with $m$ fixed). We illustrate the geometric distribution of the second largest eigenvalues in the complex plane for small $m$ and $N$, and propose a conjecture concerning their location in general. Finally, we give examples of other interval maps $f$ for which composition with permutations produces different behaviour than that obtained from the stretch-and-fold map.
We prove a fiberwise almost sure invariance principle for random piecewise expanding transformations in one and higher dimensions using recent developments on martingale techniques.
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 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.
We provide the large deviation principle for higher dimensional piecewise expanding maps and by using the functional approach of Hennion and Hervé, slightly modified.
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 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 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.
For a map $T \colon [0,1] \to [0,1]$ with an invariant measure $μ$, we study, for a $μ$-typical $x$, the set of points $y$ such that the inequality $|T^n x - y| < r_n$ is satisfied for infinitely many $n$. We give a formula for the Hausdorff dimension of this set, under the assumption that $T$ is piecewise expanding and $μ_ϕ$ is a Gibbs measure. In some cases we also show that the set has a large intersection property.
Let $f : [0, 1] \to [0, 1]$ be a piecewise expanding unimodal map of class $C^{k+1}$, with $k \geq 1$, and $μ= ρdx$ the (unique) SRB measure associated to it. We study the regularity of $ρ$. In particular, points $\mathcal{N}$ where $ρ$ is not differentiable has zero Hausdorff dimension, but is uncountable if the critical orbit of $f$ is dense. This improves on a work of Szewc (1984). We also obtain results about higher orders of differentiability of $ρ$ in the sense of Whitney.
We study the statistical properties of piecewise expanding maps in the general setting of metric measure spaces. We provide sufficient conditions for exponential mixing of such systems with explicit estimates on the constants. We also provide sufficient conditions for the existence of inducing schemes where the base transformation is Gibbs-Markov and the return times have exponential tails. Such structures can then be used to deduce finer statistical properties.
We provide a simplified proof of the existence, under some assumptions, of a spectral gap for the Perron-Frobenius operator of piecewise uniformly expanding maps on Riemannian manifolds when acting on some Sobolev spaces. Its consequences include, among others, the existence of invariant physical measures, and an exponential decay of correlations for suitable observables. These features are then adapted to different function spaces (functions with bounded variation or bounded oscillation), so as to give a new insight of - and generalize - earlier results.
We consider the regularity of measurable solutions $χ$ to the cohomological equation \[ ϕ= χ\circ T -χ, \] where $(T,X,μ)$ is a dynamical system and $ϕ\colon X\rightarrow \R$ is a $C^k$ valued cocycle in the setting in which $T \colon X\rightarrow X$ is a piecewise $C^k$ Gibbs--Markov map, an affine $β$-transformation of the unit interval or more generally a piecewise $C^{k}$ uniformly expanding map of an interval. We show that under mild assumptions, bounded solutions $χ$ possess $C^k$ versions. In particular we show that if $(T,X,μ)$ is a $β$-transformation then $χ$ has a $C^k$ version, thus…
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…
In the space of C^k piecewise expanding unimodal maps, k>=1, we characterize the C^1 smooth families of maps where the topological dynamics does not change (the "smooth deformations") as the families tangent to a continuous distribution of codimension-one subspaces (the "horizontal" directions) in that space. Furthermore such codimension-one subspaces are defined as the kernels of an explicit class of linear functionals. As a consequence we show the existence of C^{k-1+Lip} deformations tangent to every given C^k horizontal direction, for k>=2.
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.
Consider a piecewise smooth expanding map of the interval possessing several invariant subintervals and the same number of ergodic absolutely continuous invariant probability measures (ACIMs). After this system is perturbed to make the subintervals lose their invariance in such a way that there is a unique ACIM, we show how to approximate the diffusion coefficient for an observable of bounded variation by the diffusion coefficient of a related continuous time Markov chain.
For an integer $m \geq 2$, let $\mathcal{P}_m$ be the partition of the unit interval $I$ into $m$ equal subintervals, and let $\mathcal{F}_m$ be the class of piecewise linear maps on $I$ with constant slope $\pm m$ on each element of $\mathcal{P}_m$. We investigate the effect on mixing properties when $f \in \mathcal{F}_m$ is composed with the interval exchange map given by a permutation $σ\in S_N$ interchanging the $N$ subintervals of $\mathcal{P}_N$. This extends the work in a previous paper [N.P. Byott, M. Holland and Y. Zhang, DCDS, {\bf 33}, (2013) 3365--3390], where we considered only th…
The average R(t) of a smooth function with respect to the SRB measure of a smooth one-parameter family f_t of piecewise expanding interval maps is not always Lipschitz. We prove that if f_t is tangent to the topological class of f_0, then R(t) is differentiable at zero, and the derivative coincides with the resummation previously proposed by the first named author of the (a priori divergent) series given by Ruelle's conjecture.
We investigate the flexibility of the entropy (topological and metric) for the class of piecewise expanding unimodal maps. We show that the only restrictions for the values of the topological and metric entropies in this class are that both are positive, the topological entropy is at most $\log 2$, and by the Variational Principle, the metric entropy is not larger than the topological entropy. In order to have a better control on the metric entropy, we work mainly with topologically mixing piecewise expanding skew tent maps, for which there are only 2 different slopes. For those maps, there is…
Following the pioneering work of Iommi-Kiwi and Jenkinson-Pollicott-Vytnova, we continue to study the inflection points of the Lyapunov spectrum in this work. We prove that for any 3-branch piecewise linear expanding map on an interval, the number of its Lyapunov inflections is bounded above by 2. Then we continue to show that, there is a 4-branch piecewise linear expanding map, such that its Lyapunov spectrum has exactly 4 inflection points. These results give an answer to a question by Jenkinson-Pollicott-Vytnova on the least number of branches needed to observe 4 inflections in the Lyapunov…
Let $f_t:[0,1] \to [0,1]$ be a family of piecewise expanding unimodal maps with a common critical point that is dense for almost all $t \in [a,b]$. If $μ_t$ is the corresponding SRB measure for $f_t$, we study the regularity of $Γ(t)=\int ϕdμ_t$ when assuming that the family is transversal to the topological classes of these maps, more precisely, we prove that if $J_t(c)=\sum_{k=0}^{\infty} \frac{v_t(f_t^k(c))}{Df_t^k(f_t(c))} \neq 0$ for all $t$, where $v_t(x)=\partial_s f_s(x)|_{s=t}$, then $Γ(t)$ is not Lipschitz for almost all $t\in [a,b]$. Furthermore, we give the exact modulus of continu…
Recently, there has been an increasing interest on nonautonomous composition of perturbed hyperbolic systems: composing perturbations of a given hyperbolic map $F$ results in statistical behaviour close to that of $F$. We show this fact in the case of piecewise regular expanding maps. In particular, we impose conditions on perturbations of this class of maps that include situations slightly more general than what has been considered so far, and prove that these are stochastically stable in the usual sense. We then prove that the evolution of a given distribution of mass under composition of ti…
The unpredictability of chaotic nonlinear dynamics leads naturally to statistical descriptions, including probabilistic limit laws such as the central limit theorem and large deviation principle. A key tool in the Nagaev-Guivarc'h spectral method for establishing statistical limit theorems is a "twisted" transfer operator. In the abstract setting of Keller-Liverani we prove that derivatives of all orders of the leading eigenvalues and eigenprojections of the twisted transfer operators with respect to the twist parameter are stable when subjected to a broad class of perturbations. As a result, …
We give two new proofs that the SRB measure of a C^2 path f_t of unimodal piecewise expanding C^3 maps is differentiable at 0 if f_t is tangent to the topological class of f_0. The arguments are more conceptual than the one in our previous paper, but require proving Holder continuity of the infinitesimal conjugacy (a new result, of independent interest) and using spaces of bounded p-variation. The first new proof gives differentiability of higher order if f_t is smooth enough and stays in the topological class of f_0 and if the observable smooth enough (a new result). In addition, this proof d…
In this paper we show that the piecewise linear map f(x) = px for x in [0,1/p], and sx-s/p for x in (1/p,1], p > 1, 0 < s < 1 which has an expanding, onto branch and a contracting branch is eventually piecewise expanding and exact.
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 study the susceptibility function Psi(z) associated to the perturbation f_t=f+tX of a piecewise expanding interval map f. The analysis is based on a spectral description of transfer operators. It gives in particular sufficient conditions which guarantee that Psi(z) is holomorphic in a disc of larger than one. Although Psi(1) is the formal derivative of the SRB measure of f_t with respect to t, we present examples satisfying our conditions so that the SRB measure is not Lipschitz.*We propose a new version of Ruelle's conjectures.* In v2, we corrected a few minor mistakes and added Conjecture…
We show how the existence of three objects, $Ω_{\rm trap}$, ${\bf W}$, and $C$, for a continuous piecewise-linear map $f$ on $\mathbb{R}^N$, implies that $f$ has a topological attractor with a positive Lyapunov exponent. First, $Ω_{\rm trap} \subset \mathbb{R}^N$ is trapping region for $f$. Second, ${\bf W}$ is a finite set of words that encodes the forward orbits of all points in $Ω_{\rm trap}$. Finally, $C \subset T \mathbb{R}^N$ is an invariant expanding cone for derivatives of compositions of $f$ formed by the words in ${\bf W}$. We develop an algorithm that identifies these objects for tw…
Let $I\subset\mathbb{R}$ be an interval and $T_a:[0,1]\to[0,1]$, $a\in I$, a one-parameter family of piecewise expanding maps such that for each $a\in I$ the map $T_a$ admits a unique absolutely continuous invariant probability measure $μ_a$. We establish sufficient conditions on such a one-parameter family such that a given point $x\in[0,1]$ is typical for $μ_a$ for a full Lebesgue measure set of parameters $a$, i.e. $$ \frac{1}{n}\sum_{i=0}^{n-1}δ_{T_a^i(x)} \overset{\text{weak-}*}{\longrightarrow}μ_a,\qquad\text{as} n\to\infty, $$ for Lebesgue almost every $a\in I$. In particular, we consid…
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 …
I introduce Banach spaces on which it is possible to precisely characterize the spectrum of the transfer operator associated to a piecewise expanding map with Hölder weight.
We propose a systematic approach to the construction of invariant union of polytopes (IUP) in expanding piecewise affine mappings whose linear components are isotropic scalings. The approach relies on using empirical information embedded in trajectories in order to infer, and then to solve, a so-called conditioning problem for some generating collection of polytopes. A conditioning problem consists of a series of requirements on the polytopes' localisation and on the dynamical transitions between these elements. The core element of the approach is a reformulation of the problem as a set of pie…
We consider the susceptibility function Psi(z) of a piecewise expanding unimodal interval map f with unique acim mu, a perturbation X, and an observable phi. Combining previous results (deduced from spectral properties of Ruelle transfer operators) with recent work of Breuer-Simon (based on techniques from the spectral theory of Jacobi matrices and a classical paper of Agmon), we show that density of the postcritical orbit (a generic condition) implies that Psi(z) has a strong natural boundary on the unit circle. The Breuer-Simon method provides uncountably many candidates for the outer functi…
We associate to a perturbation $(f_t)$ of a (stably mixing) piecewise expanding unimodal map $f_0$ a two-variable fractional susceptibility function $Ψ_ϕ(η, z)$, depending also on a bounded observable $ϕ$. For fixed $η\in (0,1)$, we show that the function $Ψ_ϕ(η, z)$ is holomorphic in a disc $D_η\subset \mathbb{C}$ centered at zero of radius $>1$, and that $Ψ_ϕ(η, 1)$ is the Marchaud fractional derivative of order $η$ of the function $t\mapsto \mathcal{R}_ϕ(t):=\int ϕ(x)\, dμ_t$, at $t=0$, where $μ_t$ is the unique absolutely continuous invariant probability measure of $f_t$. In addition, we s…
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 consider polygonal billiards with collisions contracting the reflection angle towards the normal to the boundary of the table. In previous work, we proved that such billiards has a finite number of ergodic SRB measures supported on hyperbolic generalized attractors. Here we study the relation of these measures with the ergodic absolutely continuous invariant probability measures (acips) of the slap map, the 1-dimensional map obtained from the billiard map when the angle of reflection is always equal to zero. Our main result states that for a generic polygon, if the reflection law has a Lips…
I provide a proof of the existence of absolutely continuous invariant measures (and study their statistical properties) for multidimensional piecewise expanding systems with not necessarily bounded derivative or distortion. The proof uses basic properties of multidimensional BV functions.
We introduce a family of Banach spaces of measures, each containing the set of measures with density of bounded variation. These spaces are suitable for the study of weighted transfer operators of piecewise-smooth maps of the interval where the weighting used in the transfer operator is not better than piecewise Hölder continuous and the partition on which the map is continuous may possess a countable number of elements. For such weighted transfer operators we give upper bounds for both the spectral radius and for the essential spectral radius.
ACIM instability of piecewise expanding maps through the lens of metastability
Budapest University of Technology and Economics（布达佩斯技术与经济大学）；HUN-REN–BME Stochastics Research Group, Budapest University of Technology and Economics（匈牙利研究网络-布达佩斯技术与经济大学随机过程研究组）
We consider two classes of piecewise expanding maps $T$ of $[0,1]$: a class of uniformly expanding maps for which the Perron-Frobenius operator has a spectral gap in the space of bounded variation functions, and a class of expanding maps with a neutral fixed point at zero. In both cases, we give a large class of unbounded functions $f$ for which the partial sums of $f\circ T^i$ satisfy an almost sure invariance principle. This class contains piecewise monotonic functions (with a finite number of branches) such that: - For uniformly expanding maps, they are square integrable with respect to the…
Chaotic dynamics with sensitive dependence on initial conditions may result in exponential decay of correlation functions. We show that for one-dimensional interval maps the corresponding quantities, that is, Lyapunov exponents and exponential decay rates are related. For piecewise linear expanding Markov maps observed via piecewise analytic functions we provide explicit bounds of the decay rate in terms of the Lyapunov exponent. In addition, we comment on similar relations for general piecewise smooth expanding maps.
Two-dimensional mappings obtained by coupling two piecewise increasing expanding maps are considered. Their dynamics is described when the coupling parameter increases in the expanding domain. By introducing a coding and by analysing an admissibility condition, approximations of the corresponding symbolic systems are obtained. These approximations imply that the topological entropy is located between two decreasing step functions of the coupling parameter. The analysis firstly applies to mappings with piecewise affine local maps which allow explicit expressions and, in a second step, is extend…
Let $L:[0,1]\setminus\{d\}\rightarrow [0,1]$ be a one-dimensional Lorenz like expanding map ($d$ is the point of discontinuity), $\mathcal{P}=\{ (0,d),(d,1) \}$ be a partition of $[0,1]$ and $C^α([0,1],\mathcal{P})$ the set of piecewise Hölder-continuous potential of [0,1] with the usual $\mathcal{C}^0$ topology. In this context, we prove, improving a result of \cite{BS03}, that piecewise Hölder-continuous potential $ϕ$ satisfying \linebreak $\max\left\{ \limsup_{n \rightarrow \infty}\frac{1}{n}(S_{n}ϕ)(0),\limsup_{n \rightarrow \infty}\frac{1}{n}(S_{n}ϕ)(1)\right\}<P_{\text{top}}(ϕ,T)$ suppor…
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…
This paper generalizes the results of [13] and then provides an interesting example. We construct a family of $W$-like maps $\{W_a\}$ with a turning fixed point having slope $s_1$ on one side and $-s_2$ on the other. Each $W_a$ has an absolutely continuous invariant measure $μ_a$. Depending on whether $\frac{1}{s_1}+\frac{1}{s_2}$ is larger, equal or smaller than 1, we show that the limit of $μ_a$ is a singular measure, a combination of singular and absolutely continuous measure or an absolutely continuous measure, respectively. It is known that the invariant density of a single piecewise expa…
We study parametrised families of piecewise expanding interval mappings $T_a \colon [0,1] \to [0,1]$ with absolutely continuous invariant measures $μ_a$ and give sufficient conditions for a point $X(a)$ to be typical with respect to $(T_a, μ_a)$ for almost all parameters $a$. This is similar to a result by D. Schnellmann, but with different assumptions.
Often topological classes of one-dimensional dynamical systems are finite codimension smooth manifolds. We describe a method to prove this sort of statement that we believe can be applied in many settings. In this work we will implement it for piecewise expanding maps. The most important step will be the identification of infinitesimal deformations with primitives of Birkhoff sums (up to addition of a Lipschitz function), that allows us to use the ergodic properties of piecewise expanding maps to study the regularity of infinitesimal deformations.
We show that for almost every map in a transversal one-parameter family of piecewise expanding unimodal maps the Birkhoff sum of suitable observables along the forward orbit of the turning point satisfies the law of iterated logarithm. This result will follow from an almost sure invariance principle for the Birkhoff sum, as a function on the parameter space. Furthermore, we obtain a similar result for general one-parameter families of piecewise expanding maps on the interval.
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 consider the dynamics of `nonlinear tent maps': piecewise smooth unimodal maps with nowhere vanishing derivative. We show that if a nonlinear tent map $f$ is not infinitely renormalizable, then all its periodic orbits of sufficiently high period are hyperbolic repelling. If additionally all periodic orbits of $f$ are hyperbolic, then $f$ has at most finitely many periodic attractors and there is a hyperbolic expansion outside the basins of these periodic attractors. In particular, if a nonlinear tent map $f$ is not infinitely renormalizable and all its periodic orbits are hyperbolic repelli…
