Special $α$-limit sets ($sα$-limit sets) combine together all accumulation points of all backward orbit branches of a point $x$ under a noninvertible map. The most important question about them is whether or not they are closed. We challenge the notion of $sα$-limit sets as backward attractors for interval maps by showing that they need not be closed. This disproves a conjecture by Kolyada, Misiurewicz, and Snoha. We give a criterion in terms of Xiong's attracting center that completely characterizes which interval maps have all $sα$-limit sets closed, and we show that our criterion is satisfied in the piecewise monotone case. We apply Blokh's models of solenoidal and basic $ω$-limit sets to solve four additional conjectures by Kolyada, Misiurewicz, and Snoha relating topological properties of $sα$-limit sets to the dynamics within them. For example, we show that the isolated points in a $sα$-limit set of an interval map are always periodic, the non-degenerate components are the union of one or two transitive cycles of intervals, and the rest of the $sα$-limit set is nowhere dense. Moreover, we show that $sα$-limit sets in the interval are always both $F_σ$ and $G_δ$. Finally, since $sα$-limit sets need not be closed, we propose a new notion of $β$-limit sets to serve as backward attractors. The $β$-limit set of $x$ is the smallest closed set to which all backward orbit branches of $x$ converge, and it coincides with the closure of the $sα$-limit set. At the end of the paper we suggest several new problems about backward attractors.
We use the concept of Baire Ergodicity and Ergodic Formalism introduced to study topological and statistical attractors for interval maps, even with discontinuities. For that we also analyze the {\em wandering intervals attractors}. As a result, we establish the finiteness of the non-periodic topological attractors for piecewise $C^2$ maps with discontinuities. For $C^2$ interval maps without discontinuities, we show the coincidence of the statistical attractors with the topological ones and we calculate the upper Birkhoff averages of continuous functions for generic points, even when the map …
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.
For a non-flat $C^3$ unimodal map with a Cantor attractor, we show that for any open cover $\mathcal U$ of this attractor, the complexity function $p(\mathcal U, n)$ is of order $n\log n$. In the appendix, we construct a non-renormalizable map with a Cantor attractor for which $p(\mathcal{U}, n)$ is bounded from above for any open cover $\mathcal{U}$.
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.
Every quasi-attractor of an iterated function system (IFS) of continuous functions on a first-countable Hausdorff topological space is renderable by the probabilistic chaos game. By contrast, we prove that the backward minimality is a necessary condition to get the deterministic chaos game. As a consequence, we obtain that an IFS of homeomorphisms of the circle is renderable by the deterministic chaos game if and only if it is forward and backward minimal. This result provides examples of attractors (a forward but no backward minimal IFS on the circle) that are not renderable by the determinis…
While the forward trajectory of a point in a discrete dynamical system is always unique, in general a point can have infinitely many backward trajectories. The union of the limit points of all backward trajectories through $x$ was called by M.~Hero the "special $α$-limit" ($sα$-limit for short) of $x$. In this article we show that there is a hierarchy of $sα$-limits of points under iterations of a S-unimodal map: the size of the $sα$-limit of a point increases monotonically as the point gets closer and closer to the attractor. The $sα$-limit of any point of the attractor is the whole non-wande…
We provide an example of Cherry flow (i.e. smooth flow on the $2$-dimensional torus with a sink and a saddle) having quasi-minimal set which is an attractor. The first return map for such a flow, constructed also in the paper, is a smooth circle map having a flat interval and a non-trivial wandering interval.
Piecewise contracting maps on the interval: Hausdorff dimension, entropy and attractors
Escuela de Ingeniería, Facultad de Ingeniería y Empresa – Universidad Católica Silva Henríquez（天主教希尔瓦·恩里克兹大学工程学院、工商学院）；Department of Engineering Mathematics – University of Bristol（布里斯托大学工程数学系）
This article concerns the long-term random dynamics in regular spaces for a non-autonomous Navier-Stokes equation defined on a bounded smooth domain $\mathcal{O}$ driven by multiplicative and additive noise. For the two kinds of noise driven equations, we demonstrate the existence of a unique pullback attractor which is backward compact and asymptotically autonomous in $\mathbb{L}^2(\mathcal{O})$ and $\mathbb{H}_0^1(\mathcal{O})$, respectively. The backward-uniform flattening property of the solution is used to prove the backward-uniform pullback asymptotic compactness of the non-autonomous ra…
We study the asymptotic dynamics of piecewise contracting maps defined on a compact interval. For maps that are not necessarily injective, but have a finite number of local extrema and discontinuity points, we prove the existence of a decomposition of the support of the asymptotic dynamics into a finite number of minimal components. Each component is either a periodic orbit or a minimal Cantor set and such that the $ω$-limit set of (almost) every point in the interval is exactly one of these components. Moreover, we show that each component is the $ω$-limit set, or the closure of the orbit, of…
The aim of this paper is to prove the existence and qualitative property of random attractors for a stochastic nonlocal delayed reaction-diffusion equation (SNDRDE) on a semi-infinite interval with a Dirichlet boundary condition on the finite end. This equation models the spatial-temporal evolution of the mature individuals for a two-stage species whose juvenile and adults both diffuse that lives on a semi-infinite domain and subject to random perturbations. By transforming the SNDRDE into a random evolution equation with delay, by means of a stationary conjugate transformation, we first estab…
We show that there exist real quadratic maps of the interval whose attractors are computationally intractable. This is the first known class of such natural examples.
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 show that if $f$ is an annular homeomorphism admitting an attractor which is an irreducible annular continua with two different rotation numbers, then the entropy of $f$ is positive. Further, the entropy is shown to be associated to a $C^0$-robust rotational horseshoe. On the other hand, we construct examples of annular homeomorphisms with such attractors so that the rotation interval is uniformly large but the entropy approaches zero as much as desired. The developed techniques allow us to obtain similar results in the context of Birkhoff attractors.
Piecewise Translations is a class of dynamical systems which arises from some applications in computer science, machine learning, and electrical engineering. In dimension 1 it can also be viewed as a non-invertible generalization of Interval Exchange Transformations. These dynamical systems still possess some features of Interval Exchanges but the total volume is no longer preserved and allowed to decay. Every Piecewise Translation has a well-defined attracting subset which is the locus of our interest. We prove some results about how fast the dynamics lock onto the attractor, geometry of the …
We consider dynamical systems given by interval maps with a finite number of turning points (including critical points, discontinuities) possibly of different critical orders from two sides. If such a map $f$ is continuous and piecewise $C^2$, satisfying negative Schwarzian derivative and some summability conditions on the growth of derivatives and recurrence along the turning orbits, then $f$ has finitely many attractors whose union of basins of attraction has total probability, and each attractor supports an absolutely continuous invariant probability measure $μ$. Over each attractor there e…
We describe the behavior of solutions of switched systems with multiple globally exponentially stable equilibria. We introduce an ideal attractor and show that the solutions of the switched system stay in any given $\varepsilon$-inflation of the ideal attractor if the frequency of switchings is slower than a suitable dwell time $T$. In addition, we give conditions to ensure that the $\varepsilon$-inflation is a global attractor. Finally, we investigate the effect of the increase of the number of switchings on the total time that the solutions need to go from one region to another.
A one-dimensional confined Nonlinear Random Walk is a tuple of $N$ diffeomorphisms of the unit interval driven by a probabilistic Markov chain. For generic such walks, we obtain a geometric characterization of their ergodic stationary measures and prove that all of them have negative Lyapunov exponents. These measures appear to be probabilistic manifestations of physical measures for certain deterministic dynamical systems. These systems are step skew products over transitive subshifts of finite type (topological Markov chains) with the unit interval fiber. For such skew products, we show ther…
Observable Dynamics and the Generic Coincidence of Milnor, Statistical, and Physical Attractors
AGH University of Krakow, Faculty of Applied Mathematics（克拉科夫AGH科技大学应用数学学院）；Silesian University in Opava, Mathematical Institute（奥帕瓦西里西亚大学数学研究所）；University of Ostrava（俄斯特拉发大学）
We propose an approach to the attractors of skew products that tries to avoid unnecessary structures on the base space and rejects the assumption on the invariance of an attractor. When nonivertible maps in the base are allowed, one can encounter the mystery of the vanishing attractor. In the second part of the paper, we show that if the fiber maps are concave interval maps then contraction in the fibers does not depend on the map in the base.
We show how a parameterized family of maps of the spine of a manifold can be used to construct a family of homeomorphisms of the ambient manifold which have the inverse limits of the spine maps as global attractors. We describe applications to unimodal families of interval maps, to rotation sets, and to the standard family of circle maps.
Since the proof, at the end of the 80's, of the finiteness of the number of attractors for $C^3$ maps of the interval having negative Schwarzian derivative, it has been generally considered that the same result could be true for maps with discontinuities. In the present paper we show that this is indeed the case.
In this paper we give an elementary treatment of the dynamics of skew tent maps. We divide the two-parameter space into six regions. Two of these regions are further subdivided into infinitely many regions. All of the regions are given explicitly. We find the attractor in each subregion, determine whether the attractor is a periodic orbit or is chaotic, and also determine the asymptotic fate of every point. We find that when the attractor is chaotic, it is either a single interval or the disjoint union of a finite number of intervals; when it is a periodic orbit, all periods are possible. Some…
In diverse physical systems stable oscillatory solutions devolve into more complicated dynamical behaviour through border-collision bifurcations. Mathematically these occur when a stable fixed point of a piecewise-smooth map collides with a switching manifold as parameters are varied. The purpose of this paper is to highlight the extreme complexity possible in the subsequent dynamics. We perturb instances of the border-collision normal form in $n \ge 2$ dimensions for which the $n^{\rm th}$ iterate is a direct product of identical skew tent maps that have chaotic attractors comprised of $k \ge…
A generalized attracting horseshoe is introduced as a new paradigm for describing chaotic strange attractors (of arbitrary finite rank) for smooth and piecewise smooth maps f from Q to Q, where Q is a homeomorph of the unit interval in real m-space for any integer m > 1. The main theorems for generalized attracting horseshoes are shown to apply to Henon and Lozi maps, thereby leading to rather simple new chaotic strange attractor existence proofs that apply to a range of parameter values that includes those of earlier proofs.
We analyze a multiparameter periodically-forced dynamical system inspired in the SIR endemic model. We show that the condition on the \emph{basic reproduction number} $\mathcal{R}_0 < 1$ is not sufficient to guarantee the elimination of \emph{Infectious} individuals due to a \emph{backward bifurcation}. Using the theory of rank-one attractors, for an open subset in the space of parameters where $\mathcal{R}_0<1$, the flow exhibits \emph{persistent strange attractors}. These sets are not confined to a tubular neighbourhood in the phase space, are numerically observable and shadow the ghost of a…
We study one-parameter families of quasi-periodically forced monotone interval maps and provide sufficient conditions for the existence of a parameter at which the respective system possesses a non-uniformly hyperbolic attractor. This is equivalent to the existence of a sink-source orbit, that is, an orbit with positive Lyapunov exponent both forwards and backwards in time. The attractor itself is a non-continuous invariant graph with negative Lyapunov exponent, often referred to as "SNA". In contrast to former results in this direction, our conditions are C^2-open in the fibre maps. By applyi…
In this note we study the multifractal spectrum of Lyapunov exponents for interval maps with infinitely many branches and a parabolic fixed point. It turns out that, in strong contrast with the hyperbolic case, the domain of the spectrum is unbounded and points of non-differentiability might exist. Moreover, the spectrum is not concave. We establish conditions that ensure the existence of inflection points. We also study the thermodynamic formalism for such maps. We prove that the pressure function is real analytic in a certain interval and then it becomes equal to zero.
Skew product systems with monotone one-dimensional fibre maps driven by piecewise expanding Markov interval maps may show the phenomenon of intermingled basins. To quantify the degree of intermingledness the uncertainty exponent and the stability index were suggested by various authors and characterized (partially). Here we present an approach to evaluate/estimate these two quantities rigorously using thermodynamic formalism for the driving Markov map.
We consider semigroup actions on the unit interval generated by strictly increasing $C^r$-maps. We assume that one of the generators has a pair of fixed points, one attracting and one repelling, and a heteroclinic orbit that connects the repeller and attractor, and the other generators form a robust blender, which can bring the points from a small neighborhood of the attractor to an arbitrarily small neighborhood of the repeller. This is a model setting for partially hyperbolic systems with one central direction. We show that, under additional conditions on the non-linearity and the Schwarzian…
We calculate rigorous bounds on the Hausdorff dimension of the attractor at the accumulation of the period-doubling cascade for families of maps with quadratic, cubic, and quartic critical point. To do this, we express the attractors as the limit sets of appropriate Iterated Function Systems constructed using rigorous bounds on the corresponding renormalisation fixed point functions. We use interval arithmetic with rigorous directed rounding modes to show that the respective dimensions lie in subintervals of the intervals (0.5370,0.5392), (0.6040,0.6091), and (0.6395,0.6474).
In this paper we consider a class of skew products over transitive subshifts of finite type with interval fibers. For a natural class of 1-parameter families we prove that for all but countably many parameter values the nonwandering set (in particular, the union of all attractors and repellers) has zero measure. As a consequence, the same holds for a residual subset of the space of skew products.
Let $D$ be the set of $β\in (1, 2]$ such that $f_β$ is a symmetric tent map with finite critical orbit. For $β\in D$, by operating Denjoy like surgery on $f_β$, we constructed a $C^1$ unimodal map $\tilde{g}_β$ admitting a thick hyperbolic repelling invariant Cantor set which contains a wild Cantor attractor. The smoothness of $\tilde{g}_β$ is ensured by the effective estimation of the preimages of the critical point as well as the prescribed lengths of the inserted intervals. Furthermore, $D$ is dense in $(1, 2]$, and $\tilde{g}_β$ can not be $C^{1+α}$ because the hyperbolic repelling invaria…
In this paper, we deal with random attractors for dynamical systems forced by a deterministic noise. These kind of systems are modeled as skew products where the dynamics of the forcing process are described by the base transformation. Here, we consider skew products over the Bernoulli shift with the unit interval fiber. We study the geometric structure of maximal attractors, the orbit stability and stability of mixing of these skew products under random perturbations of the fiber maps. We show that there exists an open set $\mathcal{U}$ in the space of such skew products so that any skew prod…
We investigate the prevalence of Li-Yorke pairs for $C^2$ and $C^3$ multimodal maps $f$ with non-flat critical points. We show that every measurable scrambled set has zero Lebesgue measure and that all strongly wandering sets have zero Lebesgue measure, as does the set of pairs of asymptotic (but not asymptotically periodic) points. If $f$ is topologically mixing and has no Cantor attractor, then typical (w.r.t. two-dimensional Lebesgue measure) pairs are Li-Yorke; if additionally $f$ admits an absolutely continuous invariant probability measure (acip), then typical pairs have a dense orbit fo…
Previously published admissibility conditions for an element of $\{0,1\}^{\mathbb{Z}}$ to be the itinerary of a point of the inverse limit of a tent map are expressed in terms of forward orbits. We give necessary and sufficient conditions in terms of backward orbits, which is more natural for inverse limits. These backward admissibility conditions are not symmetric versions of the forward ones: in particular, the maximum backward itinerary which can be realised by a tent map mode locks on intervals of kneading sequences.
Special $α$-limit sets in the hyperspace of continua: closedness and entropy for interval maps
Faculty of Science, University of Split（斯普利特大学理学院）；Centre of Excellence IT4Innovations - Institute for Research and Applications of Fuzzy Modeling, University of Ostrava（俄斯特拉发大学IT4Innovations卓越中心模糊建模研究与应研究所）
For piecewise $C^1$ interval maps possibly containing critical points and discontinuities with negative Schwarzian derivative, under two summability conditions on the growth of the derivative and recurrence along critical orbits, we prove the nonexistence of wandering intervals, the existence of absolutely continuous invariant measures, and the bounded backward contraction property. The proofs are based on the method of proving the existence of absolutely continuous invariant measures of unimodal map, developed by Nowicki and van Strien.
The existence of non-continuous invariant graphs (or strange non-chaotic attractors) in quasiperiodically forced systems has generated great interest, but there are still very few rigorous results about the properties of these objects. In particular, it is not known whether the topological closure of such graphs is typically a filled-in set, i.e consists of a single interval on every fibre, or not. We give a positive answer to this question for the class of so-called pinched skew products, where non-continuous invariant graphs occur generically, provided the rotation number on the base is diop…
The tent map family is arguably the simplest 1-parametric family of maps with non-trivial dynamics and it is still an active subject of research. In recent works the second author, jointly with J. Yorke, studied the graph and backward limits of S-unimodal maps. In this article we generalize those results to tent-like unimodal maps. By tent-like here we mean maps that share fundamental properties that characterize tent maps, namely unimodal maps without wandering intervals nor attracting cycles and whose graph has a finite number of nodes.
Intermingled basins: Kan's example on the Riemann sphere
Shahid Beheshti University（沙希德贝赫什提大学）；KdV Institute for Mathematics, University of Amsterdam（阿姆斯特丹大学KdV数学研究所）；Mathematical Institute, University of Leiden（莱顿大学数学研究所）；Department of Mathematics, Imperial College London（伦敦帝国理工学院数学系）
We study the asymptotic dynamics of maps which are piecewise contracting on a compact space. These maps are Lipschitz continuous, with Lipschitz constant smaller than one, when restricted to any piece of a finite and dense union of disjoint open pieces. We focus on the topological and the dynamical properties of the (global) attractor of the orbits that remain in this union. As a starting point, we show that the attractor consists of a finite set of periodic points when it does not intersect the boundary of a contraction piece, which complements similar results proved for more specific classes…
We study the topological properties of attractors of Iterated Function Systems (I.F.S.) on the real line, consisting of affine maps of homogeneous contraction ratio. These maps define what we call a second generation I.F.S.: they are uncountably many and the set of their fixed points is a Cantor set. We prove that when this latter either is the attractor of a finite, non-singular, hyperbolic, I.F.S. (of first generation), or it possesses a particular dissection property, the attractor of the second generation I.F.S. consists of finitely many closed intervals.
We extend a result regarding the Random Backward Iteration algorithm for drawing Julia sets (known to work for certain rational semigroups containing a non-Möbius element) to a class of Möbius semigroups which includes certain settings not yet been dealt with in the literature, namely, when the Julia set is not a thick attractor in the sense given in [David Fried, Sebastian. M. Marotta, and Rich Stankewitz, Complex dynamics of Möbius semigroups, Ergodic Theory Dynam. Systems, 32(6):1889--1929, 2012].
Generic continuous Lebesgue measure-preserving interval maps are nowhere monotone but invertible a.e
Czech Technical University in Prague（布拉格捷克理工大学）；University of Maribor（马里博尔大学）；Abdus Salam International Centre for Theoretical Physics (ICTP)（阿卜杜斯·萨拉姆国际理论物理中心（ICTP））；University of Ostrava（俄斯特拉发大学）；Aix Marseille Univ（艾克斯-马赛大学）
The main goal of this paper is to study topological and measure-theoretic properties of an intriguing family of strange planar attractors. Building towards these results, we first show that any generic Lebesgue measure preserving map $f$ generates the pseudo-arc as inverse limit with $f$ as a single bonding map. These maps can be realized as attractors of disc homeomorphisms in such a way that the attractors vary continuously (in Hausdorff distance on the disc) with the change of bonding map as a parameter. Furthermore, for generic Lebesgue measure preserving maps $f$ the background Oxtoby-Ula…
We establish the (level-1) large deviation principles for three kinds of means associated with the backward continued fraction expansion. We show that: for the harmonic and geometric means, the rate functions vanish exactly at one point; for the arithmetic mean, it is completely degenerate, vanishing at every point in its effective domain. Our method of proof employs the thermodynamic formalism for finite Markov shifts, and a multifractal analysis for the Rényi map generating the backward continued fraction digits. We completely determine the class of unbounded arithmetic functions for which t…
The structure of a self-similar set with open set condition does not change under magnification. For self-affine sets the situation is completely different. We consider planar self-affine Cantor sets E of the type studied by Bedford, McMullen, Gatzouras and Lalley, for which the projection onto the horizontal axis is an interval. We show that within small square neighborhoods of almost each point x in E, with respect to many product measures on address space, E is well approximated by product sets of an interval and a Cantor set. Even though E is totally disconnected, the limit sets have the p…
We establish arithmetical properties and provide essential bounds for bi-sequences of approximation coefficients associated with the natural extension of maps, leading to continued fraction-like expansions. These maps are realized as the fractional part of M$\operatorname{\ddot{o}}$bius transformations which carry the end points of the unit interval to zero and infinity, extending the classical regular and backwards continued fractions expansions.
We will establish several arithmetic and geometric properties regarding the bi-sequences of approximation coefficients (BAC) associated with the two one-parameter families of piecewise-continuous Mobius transformations introduced by Haas and Molnar. The Gauss and Renyi maps, which lead to the expansions of irrational numbers on the interval as regular and backwards continued fractions, are realized as special cases. The results are natural generalizations of theorems from Diophantine approximation.
