Mild mixing property for special flows under piecewise constant functions

Fractal dimensions of continuous piecewise linear iterated function systems

We consider iterated function systems on the real line that consist of continuous, piecewise linear functions. Under a mild separation condition, we show that the Hausdorff and box dimensions of the attractor are equal to the minimum of 1 and the exponent which comes from the most natural system of covers of the attractor.

Topological mild mixing of all orders along polynomials

A minimal system $(X,T)$ is topologically mildly mixing if all non-empty open subsets $U,V$, $\{n\in \Z: U\cap T^{-n}V\neq \emptyset\}$ is an IP$^*$-set. In this paper we show that if a minimal system is topologically mildly mixing, then it is mild mixing of all orders along polynomials. That is, suppose that $(X,T)$ is a topologically mildly mixing minimal system, $d\in \N$, $p_1(n),\ldots, p_d(n)$ are integral polynomials with no $p_i$ and no $p_i-p_j$ constant, $1\le i\neq j\le d$, then for all non-empty open subsets $U , V_1, \ldots, V_d $, $\{n\in \Z: U\cap T^{-p_1(n) }V_1\cap T^{-p_2(n)}…

Stretched-exponential mixing for $\mathscr{C}^{1+α}$ skew products with discontinuities

Consider the skew product $F:\mathbb{T}^2 \to \mathbb{T}^2$, $F(x,y)= (f(x),y+τ(x))$, where $f:\mathbb{T}^1\to \mathbb{T}^1$ is a piecewise $\mathscr{C}^{1+α}$ expanding map on a countable partition and $τ:\mathbb{T}^1 \to \mathbb{R}$ is piecewise $\mathscr{C}^1$. It is shown that if $τ$ is not Lipschitz-cohomologous to a piecewise constant function on the joint partition of $τ$ and $f$, then $F$ is mixing at a stretched-exponential rate.

Exponential Mixing for Skew Products with Discontinuities

We consider the skew product $F: (x,u) \mapsto (f(x), u + τ(x))$, where the base map $f : \mathbb{T}^{1} \to \mathbb{T}^{1}$ is piecewise $\mathcal{C}^{2}$, covering and uniformly expanding, and the fibre map $τ: \mathbb{T}^{1} \to \mathbb{R}$ is piecewise $\mathcal{C}^{2}$. We show the dichotomy that either this system mixes exponentially or $τ$ is cohomologous (via a Lipschitz function) to a piecewise constant.

On the mixing coefficients of piecewise monotonic maps

We investigate the mixing coefficients of interval maps satisfying Rychlik's conditions. A mixing Lasota-Yorke map is reverse $ϕ$-mixing. If its invariant density is uniformly bounded away from 0, it is $ϕ$-mixing iff all images of all orders are big in which case it is $ψ$-mixing. Among $\b$-transformations, non-$ϕ$-mixing is generic. In this sense, the asymmetry of $ϕ$-mixing is natural.

Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps

For a class of piecewise hyperbolic maps in two dimensions, we propose a combinatorial definition of topological entropy by counting the maximal, open, connected components of the phase space on which iterates of the map are smooth. We prove that this quantity dominates the measure theoretic entropies of all invariant probability measures of the system, and then construct an invariant measure whose entropy equals the proposed topological entropy. We prove that our measure is the unique measure of maximal entropy, that it is ergodic, gives positive measure to every open set, and has exponential…

刚性函数、IP系统与拓扑弱混合

Rigid Functions, IP-Systems, and Topological Mild Mixing

该研究通过连续可观测量探讨拓扑动力系统的一致刚性与弱混合，明确了SIP、IP相关刚性可观测量的代数结构及因子构造条件，给出弱混合的等价判定准则。

Square-mean S-asymptotically $ω$-periodic solution for a stochastic fractional evolution equation driven by Lévy noise with piecewise constant argument

In this paper, we introduce some concepts of square-mean S-asymptotically $ω$-periodic stochastic processes. Using the stochastic analysis method and the Banach contraction mapping principle, we establish the existence and uniqueness results of the mild solution and the square-mean S-asymptotically $ω$-periodic solution for a semilinear nonautonomous stochastic fractional evolution equation driven by Lévy noise.

On piecewise expanding maps

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.

On the mixing properties of piecewise expanding maps under composition with permutations

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 eigenvalu…

On convergence of linear response formulae in some piecewise hyperbolic maps

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…

Constant Slope Maps on the Extended Real Line

For a transitive countably piecewise monotone Markov interval map we consider the question whether there exists a conjugate map of constant slope. The answer varies depending on whether the map is continuous or only piecewise continuous, whether it is mixing or not, what slope we consider, and whether the conjugate map is defined on a bounded interval, half-line or the whole real line (with the infinities included).

Mixing sets for non-mixing transformations

For different classes of measure preserving transformations, we investigate collections of sets that exhibit the property of lightly mixing. Lightly mixing is a stronger property than topological mixing, and requires that a lim inf is positive. In particular, we give a straightforward proof that any mildly mixing transformation T has a dense algebra C such that T is lightly mixing on C. Also, we provide a hierarchy for the properties of lightly mixing, sweeping out and uniform sweeping out for dense collections, and dense algebras of sets. As a result, stronger mixing realizations are given fo…

Phase transitions in a piecewise expanding coupled map lattice with linear nearest neighbour coupling

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.

Absence of mixing for interval translation mappings and some generalizations

We consider piecewise monotone maps, we show that an ergodic measure for which the map is invertible almost everywhere can not be mixing. It follows that every ergodic measure for an interval translation mapping is not mixing. We also show that double rotations without periodic points have an ergodic but not weakly mixing invariant measure. This article is dedicated to the memory of Anatoly Mikhailovich Stepin.

Ratner's property and mixing for special flows over two-dimensional rotations

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…

Invariant measures for interval maps with critical points and singularities

We prove that, under a mild summability condition on the growth of the derivative on critical orbits any piecewise monotone interval map possibly containing discontinuities and singularities with infinite derivative (cusp map) admits an ergodic invariant probability measures which is absolutely continuous with respect to Lebesgue measure.

$Δ$-weakly mixing subsets along a collection of sequences of integers

In this paper, we propose a mild condition, named Condition $(**)$, for collections of sequence of integers and show that for any measure preserving system the Pinsker $σ$-algebra is a characteristic $σ$-algebra for the averages along a collection satisfying Condition $(**)$. We introduce the notion of $Δ$-weakly mixing subsets along a collection of sequences of integers and show that positive topological entropy implies the existence of $Δ$-weakly mixing subsets along a collection of "good" sequences. As a consequence, we show that positive topological entropy implies multi-variant Li-Yorke c…

Open Sets of Exponentially Mixing Anosov Flows

We prove that an Anosov flow with $\mathcal{C}^{1}$ stable bundle mixes exponentially whenever the stable and unstable bundles are not jointly integrable. This allows us to show that if a flow is sufficiently close to a volume-preserving Anosov flow and $\operatorname{dim} \mathbb{E}_s = 1$, $\operatorname{dim} \mathbb{E}_u \geq 2$ then the flow mixes exponentially whenever the stable and unstable bundles are not jointly integrable.This implies the existence of non-empty open sets of exponentially mixing Anosov flows. As part of the proof of this result we show that $\mathcal{C}^{1+}$ uniforml…

偏周期解的存在性和唯一性：具有分段常数论证的微分方程

Existence and uniqueness of Remotely Almost Periodic solutions of differential equations with piecewise constant argument

本文研究了具有分段常数论证的微分方程的远程几乎周期解的存在性和唯一性，通过指数分裂和Lipschitz条件推导了充分条件，并应用于非自治Lasota-Wazewska模型。

典型Salem替换悬吊流的定量弱混合性

Quantitative weak mixing for typical Salem substitution suspension flows

本文研究了Salem替换流的定量弱混合性，证明了其弱混合性速率略逊于$\log\log$幂，并首次给出了Salem伯努利卷积的傅里叶变换的定量衰减率。

Flows with Ratner's property have discrete essential centralizer

We show that a free, ergodic action of $\mathbb{R}$ with (finite) Ratner's property has countable discrete essential centralizer. Moreover we show that such flows are mildly mixing.

随机Chirikov标准映射的定量指数混合性

Quantitative exponential mixing for the randomized Chirikov standard map

本文研究了随机Chirikov标准映射在二维环面上的混合性质，通过建立不可压缩随机动力系统准则，证明了在激励强度足够大时，随机映射具有几乎必然的定量指数混合性，并给出了更温和的参数条件以获得定性指数混合性和增强的耗散性。

The Infimum of Lipschitz Constants in the Conjugacy Class of an Interval Map

How can we interpret the infimum of Lipschitz constants in a conjugacy class of interval maps? For positive entropy maps, the exponential of the topological entropy gives a well-known lower bound. We show that for piecewise monotone interval maps as well as for $C^{\infty}$ interval maps, these two quantities are equal, but for countably piecewise monotone maps, the inequality can be strict. Moreover, in the topologically mixing and Markov case, we characterize the infimum of Lipschitz constants as the exponential of the Salama entropy of a certain reverse Markov chain associated with the map.…

Mean convergence for intermediately trimmed Birkhoff sums of observables with regularly varying tails

On a measure theoretical dynamical system with spectral gap property we consider non-integrable observables with regularly varying tails and fulfilling a mild mixing condition. We show that the normed trimmed sum process of these observables then converges in mean. This result is new also for the special case of i.i.d. random variables and contrasts the general case where mean convergence might fail even though a strong law of large numbers holds. To illuminate the required mixing condition we give an explicit example of a dynamical system fulfilling a spectral gap property and an observable w…

Ratner's property for special flows over irrational rotations under functions of bounded variation

We consider special flows over the rotation by an irrational $α$ under the roof functions of bounded variation without continuous, singular part in the Lebesgue decomposition and the sum of jumps $\neq 0$. We show that all such flows are weakly mixing. Under the additional assumption that $α$ has bounded partial quotients, we study weak Ratner's property. We establish this property whenever an additional condition (stable under sufficiently small perturbations) on the set of jumps is satisfied. While it is classical that the flows under consideration are not mixing, one more condition on the s…

On the fractional susceptibility function of piecewise expanding maps

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…