本文提出一种严格计算机算法，通过构造显式离散化度量来证明复多项式映射在 Julia 集邻域上扩张，从而判定其双曲性，并讨论算法改进及逼近欧几里得度量的问题。
摘要：
我们描述了一种严格的计算机算法，用于尝试构造一个显式的离散化度量，使得一个复多项式映射在其 Julia 集的给定邻域上是扩张的。我们证明构造出这样的度量即可证明该映射是双曲的。我们还考察了该算法能否被改进的问题，以及与之相关的如何构造尽可能接近欧几里得度量的度量的问题。最后，我们给出了用该算法的实现所生成的若干例子。
英文摘要：
We describe a rigorous computer algorithm for attempting to construct an explicit, discretized metric for which a complex polynomial map is expansive on a given neighborhood of its Julia set. We show construction of such a metric proves the map is hyperbolic. We also examine the question of whether the algorithm can be improved, and the related question of how to build a metric as close to euclidean as possible. Finally, we give several examples generated with our implementation of this algorithm.
Chaotic attractors in the two-dimensional border-collision normal form (a piecewise-linear map) can persist throughout open regions of parameter space. Such robust chaos has been established rigorously in some parameter regimes. Here we provide formal results for robust chaos in the original parameter regime of [S. Banerjee, J.A. Yorke, C. Grebogi, Robust Chaos, Phys. Rev. Lett. 80(14):3049--3052, 1998]. We first construct a trapping region in phase space to prove the existence of a topological attractor. We then construct an invariant expanding cone in tangent space to prove that tangent vect…
We investigate a phenomenon observed by W. Thurston wherein one constructs a pseudo-Anosov homeomorphism on the limit set of a certain lift of a piecewise-linear expanding interval map. We reconcile this construction with a special subclass of generalized pseudo-Anosovs, first defined by de Carvalho. From there we classify the circumstances under which this construction produces a pseudo-Anosov. As an application, we produce for each $g \geq 1$ a pseudo-Anosov $ϕ_g$ on the surface of genus $g$ that preserves an algebraically primitive translation structure and whose dilatation $λ_g$ is a Salem…
The aim of this paper is to show $J$-stability of immediately expanding rational maps over an algebraically closed, complete, and non-Archimedean field, which is an analogue of R. Mane, P. Sad, and D. Sullivan's theorem of $J$-stability in complex dynamical systems.
In this paper we consider Hausdorff dimension of the sets of Li-Yorke pairs for some chaotic dynamical systems including $A$-coupled expanding systems. We prove that Li-Yorke pairs of $A$- coupled-expanding system under some conditions have full hausdorff dimension on the invariant set. we generalize the result of [8] on the Hausdorff dimension of Li-Yorke pairs of dynamical systems topologically conjugate to the full shift and having a self-similar invariant set to the case of dynamical system conjugated to some kind of subshifts. And Hausdorff dimension of "chaotic invariant set" for some ki…
In this paper, we investigate annealed and quenched limit theorems for random expanding dynamical systems. Making use of functional analytic techniques and more probabilistic arguments with martingales, we prove annealed versions of a central limit theorem, a large deviation principle, a local limit theorem, and an almost sure invariance principle. We also discuss the quenched central limit theorem, dynamical Borel-Cantelli lemmas, Erdös-Rényi laws and concentration inequalities.
In order to adapt to the differentiable setting a formula for linear response proved by Pollicott and Vytnova in the analytic setting, we show a result of regularity of dynamical determinants of expanding maps of the circle. The main tool is the decomposition of a transfer operator as a sum of a nuclear part and a "small" bounded part.
We consider a class of dynamical systems, which we call weakly coarse expanding, which is a generalization to the postcritically infinite case of expanding Thurston maps as discussed by Bonk-Meyer and is closely related to coarse expanding conformal systems as defined by Haissinsky-Pilgrim. We prove existence and uniqueness of equilibrium states for a wide class of potentials, as well as statistical laws such as a central limit theorem, law of iterated logarithm, exponential decay of correlations and a large deviation principle. Further, if the system is defined on the 2-sphere, we prove all s…
We consider non-uniformly expanding maps on compact Riemannian manifolds of arbitrary dimension, possibly having discontinuities and/or critical sets, and show that under some general conditions they admit an induced Markov tower structure for which the decay of the return time function can be controlled in terms of the time generic points need to achieve some uniform expanding behavior. As a consequence we obtain some rates for the decay of correlations of those maps and conditions for the validity of the Central Limit Theorem.
We introduce a class of discrete dynamical systems that we call \emph{virtually expanding}. This is an open subset of self-covering maps on a closed manifold which contains all expanding maps and some partially hyperbolic volume-expanding maps. We show that the Perron-Frobenius operator is quasi-compact on a Sobolev space of positive order for such class of dynamical systems.
In the present paper, we study the distribution of the return points in the fibers for a RDS (random dynamical systems) nonuniformly expanding preserving an ergodic probability, we also show the abundance of nonlacunarity of hyperbolic times that are obtained along the orbits through the fibers. We conclude that any ergodic measure with positive Lyapunov exponents satisfies the nonuniform specification property between fibers. As consequences, we prove that any expanding measure is the limit of probability measure whose measures of disintegration on the fibers are supported by a finite number …
In [30] different statistical behavior of dynamical orbits without syndetic center are considered. In present paper we continue this project and consider different statistical behavior of dynamical orbits with nonempty syndetic center: Two of sixteen cases appear (for which other fourteen cases are still unknown) in transive topologically expanding or hyperbolic systems and are discovered to have full topological entropy for which it is also true if combined with non-recurrence and multifractal analysis such as quasi-regular set, irregular set and level sets. In this process a strong entropy-d…
We introduce the Markov extension, represented schematically as a tower, to the study of dynamical systems with holes. For tower maps with small holes, we prove the existence of conditionally invariant probability measures which are absolutely continuous with respect to Lebesgue measure (abbreviated a.c.c.i.m.). We develop restrictions on the Lebesgue measure of the holes and simple conditions on the dynamics of the tower which ensure existence and uniqueness in a class of Holder continuous densities. We then use these results to study the existence and properties of a.c.c.i.m. for expanding m…
This paper discusses the evolution of probability distributions for certain time-dependent dynamical systems. Exponential loss of memory is proved for expanding maps and for one-dimensional piecewise expanding maps with slowly varying parameters.
We establish almost sure invariance principles, a strong form of approximation by Brownian motion, for non-stationary time-series arising as observations on dynamical systems. Our examples include observations on sequential expanding maps, perturbed dynamical systems, non-stationary sequences of functions on hyperbolic systems as well as applications to the shrinking target problem in expanding systems.
We define iterated monodromy groups of more general structures than partial self-covering. This generalization makes it possible to define a natural notion of a combinatorial model of an expanding dynamical system. We prove that a naturally defined "Julia set" of the generalized dynamical systems depends only on the associated iterated monodromy group. We show then that the Julia set of every expanding dynamical system is an inverse limit of simplicial complexes constructed by inductive cut-and-paste rules.
Given a rational map of the Riemann sphere and a subset $A$ of its Julia set, we study the $A$-exceptional set, that is, the set of points whose orbit does not accumulate at $A$. We prove that if the topological entropy of $A$ is less than the topological entropy of the full system then the $A$-exceptional set has full topological entropy. Furthermore, if the Hausdorff dimension of $A$ is smaller than the dynamical dimension of the system then the Hausdorff dimension of the $A$-exceptional set is larger than or equal to the dynamical dimension, with equality in the particular case when the dyn…
This paper discusses the thermodynamic properties for certain time-dependent dynamical systems. In particular, we are interested in time-dependent dynamical systems with the specification property. We show that each time-dependent dynamical system given by a sequence of surjective continuous self maps of a compact metric space with the specification property has positive topological entropy and all points are entropy point. In particular, it is proved that these systems are topologically chaotic. We will treat the dynamics of uniformly Ruelle-expanding time-dependent dynamical systems on compa…
We estimate the speed of decay of correlations for general nonuniformly expanding dynamical systems, using estimates on the time the system takes to become really expanding. Our method can deal with fast decays, such as exponential or stretched exponential. We prove in particular that the correlations of the Alves-Viana map decay in $O(e^{-c \sqrt{n}})$.
The goal of this paper is to construct invariant dynamical objects for a (not necessarily invertible) smooth self map of a compact manifold. We prove a result that takes advantage of differences in rates of expansion in the terms of a sheaf cohomological long exact sequence to create unique lifts of finite dimensional invariant subspaces of one term of the sequence to invariant subspaces of the preceding term. This allows us to take invariant cohomological classes and under the right circumstances construct unique currents of a given type, including unique measures of a given type, that repres…
We consider the multifractal formalism for the dynamics of semigroups of rational maps on the Riemann sphere and random complex dynamical systems. We elaborate a multifractal analysis of level sets given by quotients of Birkhoff sums with respect to the skew product associated with a semigroup of rational maps. Applying these results, we perform a multifractal analysis of the Hölder regularity of limit state functions of random complex dynamical systems.
We prove a vector-valued almost sure invariance principle for some classes of time dependent non-uniformly distance expanding dynamical systems. The models we have in mind are certain sequential versions of the smooth non-uniformly distance expanding maps considered in \cite{castro} and \cite{Vara}, as well as certain types of sequences of covering maps. Our results rely on the theory of complex projective metrics which was developed in \cite{Rug}, together with the spectral methods of Gouëzel \cite{GO}. A big advantage in applying the theory of complex cones here is that it also yields additi…
We obtain an analogue of the prime number theorem for a class of branched covering maps on the $2$-sphere called expanding Thurston maps $f$, which are topological models of some rational maps without any smoothness or holomorphicity assumption. More precisely, by studying dynamical zeta functions and, more generally, dynamical Dirichlet series for $f$, we show that the number of primitive periodic orbits of $f$, ordered by a weight on each point induced by a non-constant (eventually) positive real-valued Hölder continuous function $ϕ$ on $S^2$ satisfying some additional regularity conditions,…
We prove results on mixing and mixing rates for toral extensions of nonuniformly expanding maps with subexponential decay of correlations. Both the finite and infinite measure settings are considered. Under a Dolgopyat-type condition on nonexistence of approximate eigenfunctions, we prove that existing results for (possibly nonMarkovian) nonuniformly expanding maps hold also for their toral extensions.
We continue the study of non-invertible topological dynamical systems with expanding behavior. We introduce the class of {\em finite type} systems which are characterized by the condition that, up to rescaling and uniformly bounded distortion, there are only finitely many iterates. We show that subhyperbolic rational maps and finite subdivision rules (in the sense of Cannon, Floyd, Kenyon, and Parry) with bounded valence and mesh going to zero are of finite type. In addition, we show that the limit dynamical system associated to a selfsimilar, contracting, recurrent, level-transitive group act…
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.
Let $f: M \to M$ be a $C^{1+α}$ map/diffeomorphism of a compact Riemannian manifold $M$ and $μ$ be an expanding/hyperbolic ergodic $f$-invariant Borel probability measure on $M$. Assume $f$ is average conformal expanding/hyperbolic on the support set $W$ of $μ$ and $W$ is locally maximal. For any subset $A\subset W$ with small entropy or dimension, we investigate the topological entropy and Hausdorff dimensions of the $A$-exceptional set and the limit $A$-exceptional set.
In this note, we obtain verifiable sufficient conditions for the extreme value distribution for a certain class of skew product extensions of non-uniformly hyperbolic base maps. We show that these conditions, formulated in terms of the decay of correlations on the product system and the measure of rapidly returning points on the base, lead to a distribution for the maximum of $Φ(p) = -\log(d(p, p_0))$ that is of the first type. In particular, we establish the Type I distribution for $S^1$ extensions of piecewise $C^2$ uniformly expanding maps of the interval, non-uniformly expanding maps of th…
We prove that for conformal expanding maps the return time does have constant multifractal spectrum. This is the counterpart of the result by Feng and Wu in the symbolic setting.
In some maps the existence of an attractor with a positive Lyapunov exponent can be proved by constructing a trapping region in phase space and an invariant expanding cone in tangent space. If this approach fails it may be possible to adapt the strategy by considering an induced map (a first return map for a well-chosen subset of phase space). In this paper we show that such a construction can be applied to the two-dimensional border-collision normal form (a continuous piecewise-linear map) if a certain set of conditions are satisfied and develop an algorithm for checking these conditions. The…
We associate with every locally expanding self-covering $f:M\to M$ of a compact path connected metric space a finitely presented group $V_f$. We prove that this group is a complete invariant of the dynamical system: two groups $V_{f_1}$ and $V_{f_2}$ are isomorphic as abstract groups if and only if the corresponding dynamical systems are topologically conjugate. We also show that the commutator subgroup of $V_f$ is simple, and give a topological interpretation of $V_f/V_f'$.
Let $T \colon M \to M$ be a nonuniformly expanding dynamical system, such as logistic or intermittent map. Let $v \colon M \to \mathbb{R}^d$ be an observable and $v_n = \sum_{k=0}^{n-1} v \circ T^k$ denote the Birkhoff sums. Given a probability measure $μ$ on $M$, we consider $v_n$ as a discrete time random process on the probability space $(M, μ)$. In smooth ergodic theory there are various natural choices of $μ$, such as the Lebesgue measure, or the absolutely continuous $T$-invariant measure. They give rise to different random processes. We investigate relation between such processes. We sh…
Given a $\mathcal{C}^\infty$ expanding map $T$ of the circle, we construct a Hilbert space $\mathcal{H}$ of smooth functions on which the transfer operator $\mathcal{L}$ associated to $T$ acts as a compact operator. This result is made quantitative (in terms of singular values of the operator $\mathcal{L}$ acting on $\mathcal{H}$) using the language of Denjoy-Carleman classes. Moreover, the nuclear power decomposition of Baladi and Tsujii can be performed on the space $\mathcal{H}$, providing a bound on the growth of the dynamical determinant associated to $\mathcal{L}$.
The purpose of this paper is twofold. In one direction, we extend the spectral method for random piecewise expanding and hyperbolic dynamics developed by the first author \textit{et al}. to establish quenched versions of the large deviation principle, central limit theorem and the local central limit theorem for \emph{vector-valued} observables. We stress that the previous works considered exclusively the case of scalar-valued observables. In another direction, we show that this method can be used to establish a variety of new limit laws (either for scalar or vector-valued observables) that ha…
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 prove quenched versions of (i) a large deviations principle (LDP), (ii) a central limit theorem (CLT), and (iii) a local central limit theorem (LCLT) for non-autonomous dynamical systems. A key advance is the extension of the spectral method, commonly used in limit laws for deterministic maps, to the general random setting. We achieve this via multiplicative ergodic theory and the development of a general framework to control the regularity of Lyapunov exponents of \emph{twisted transfer operator cocycles} with respect to a twist parameter. While some versions of the LDP and CLT have previo…
In this paper we obtain an almost sure invariance principle for convergent sequences of either Anosov diffeomorphisms or expanding maps on compact Riemannian manifolds and prove an ergodic stability result for such sequences. The sequences of maps need not correspond to typical points of a random dynamical system. The methods in the proof rely on the stability of compositions of hyperbolic dynamical systems. We introduce the notion of sequential conjugacies and prove that these vary in a Lipschitz way with respect to the generating sequences of dynamical systems. As a consequence, we prove sta…
The tent map is an elementary example of an interval map possessing many interesting properties, such as dense periodicity, exactness, Lipschitzness and a kind of length-expansiveness. It is often used in constructions of dynamical systems on the interval/trees/graphs. The purpose of the present paper is to construct, on totally regular continua (i.e. on topologically rectifiable curves), maps sharing some typical properties with the tent map. These maps will be called length-expanding Lipschitz maps, briefly LEL maps. We show that every totally regular continuum endowed with a suitable metric…
In this paper, time-dependent dynamical systems given by sequences of maps are studied. For systems built from expanding C^2-maps on a compact Riemannian manifold M with uniform bounds on expansion factors and derivatives, we provide formulas for the metric and topological entropy. If we only assume that the maps are C^1, but act in the same way on the fundamental group of M, we can show the existence of an equi-conjugacy to an autonomous system, implying a full variational principle for the entropy. Finally, we introduce the notion of strong uniform expansivity that generalizes the classical …
We study the problem of classifying stationary measures and orbit closures for non-abelian action on a surface with a given smooth invariant measure. Using a result of Brown and Rodriguez Hertz, we show that under a certain finite verifiable average growth condition, the only nonatomic stationary measure is the given smooth invariant measure, and every orbit closure is either finite or dense. Moreover, every point with infinite orbit equidistributes on the surface with respect to the smooth invariant measure. This is analogous to the results of Benoist-Quint and Eskin-Lindenstrauss in the homo…
In this paper we will prove various probabilistic limit theorems for some classes of distance expanding sequential dynamical systems (SDS). Our starting point here is certain sequential complex Ruelle-Perron-Frobenius (RPF) theorems which were proved in \cite{book} and \cite{ASIP me} using contraction properties of a complex version of the projective Hilbert metric developed in \cite{Rug}. We will start from the growth rate of the variances of the underlying partial sums. This is well understood in the random dynamics setup, when the maps are stationary, but not in the SDS setup, where various…
We consider a $\mathbb{R}$-extension of one dimensional uniformly expanding open dynamical systems and prove a new explicit estimate for the asymptotic spectral gap. To get these results, we use a new application of a "global normal form" for the dynamical system, a "semiclassical expression beyond the Ehrenfest time" that expresses the transfer operator at large time as a sum over rank one operators (each is associated to one orbit). In this paper we establish the validity of the so-called "diagonal approximation" up to twice the local Ehrenfest time.
The principal aim of this paper is to give a dynamical presentation of the Jiang-Su algebra. Originally constructed as an inductive limit of prime dimension drop algebras, the Jiang-Su algebra has gone from being a poorly understood oddity to having a prominent positive role in George Elliott's classification programme for separable, nuclear C*-algebras. Here, we exhibit an etale equivalence relation whose groupoid C*-algebra is isomorphic to the Jiang-Su algebra. The main ingredient is the construction of minimal homeomorphisms on infinite, compact metric spaces, each having the same cohomolo…
The Ising model provides a natural mapping for many computationally hard combinatorial optimization problems (COPs). Consequently, dynamical system-inspired computing models and hardware platforms that minimize the Ising Hamiltonian, have recently been proposed as a potential candidate for solving COPs, with the promise of significant performance benefit. However, the Ising model, and consequently, the corresponding dynamical system-based computational models primarily consider quadratic interactions among the nodes. Computational models considering higher order interactions among Ising spins …
