Application of Gray codes to the study of the theory of symbolic dynamics of unimodal maps

In this paper we provide a closed mathematical formulation of our previous results in the field of symbolic dynamics of unimodal maps. This being the case, we discuss the classical theory of applied symbolic dynamics for unimodal maps and its reinterpretation using Gray codes. This connection was previously emphasized but no explicit mathematical proof was provided. The work described in this paper not only contributes to the integration of the different interpretations of symbolic dynamics of unimodal maps, it also points out some inaccuracies that exist in previous works.     

Unimodal maps as boundary restrictions of two-dimensional full-folding maps

It is shown that every unimodal map is realized as a restriction of a simple map defined on the unit disc to a part of its boundary. Our two-dimensional map is called a full-folding map, which is defined generally on a compact metric space. It is a generalization of the full tent map in that it has two homeomorphic inverse maps and thus every non-critical point has two inverse images.     

Topological and metrical conditions for Collet-Eckmann unimodal maps

1. IntroductionColletEchazann maps (i.e. every ColletEckmann maP has positive lower LyaPunovexponent at the critical value) play a very important role in the study of one dimensionaldynamical systems. They have very good metricaI and ergodic properties[ll2], for example,every COllet-Eckmann map admits an absolutely cootinuous invariani m.asur.[1].In I3,4] Benedicks and Carleson proved that Collet-Eckmann maps are abundant in theso-caJled quadratic family which is a typical family of one …     

Which families of -modal maps are full?

In this paper we shall show that certain conditions which are sufficient for a family of one-dimensional maps to be full cannot be dispensed with.

     

Canonical symbolic dynamics for one-dimensional generalized solenoids

We define canonical subshift of finite type covers for Williams' one-dimensional generalized solenoids, and use resulting invariants to distinguish some closely related solenoids.

     

Computing topological entropy for periodic sequences of unimodal maps

In this paper we introduce an algorithm which allows us to compute the topological entropy of a class of piecewise monotone continuous interval maps. The algorithm can be applied to a class of economic models called duopolies, and it can be useful to compute the topological entropy of periodic sequences of continuous maps which have been used in some population growth models.     

Computation of Nielsen numbers for maps of closed surfaces

Let be a closed surface, and let be a map. We would like to determine Nielsen fixed point theory provides a lower bound for , called the Nielsen number, which is easy to define geometrically and is difficult to compute.

We improve upon an algebraic method of calculating developed by Fadell and Husseini, so that the method becomes algorithmic for orientable closed surfaces up to the distinguishing of Reidemeister orbits. Our improvement makes tractable calculations of Nielsen numbers for many maps on surfaces of negative Euler characteristic. We apply the improved method to self-maps on the connected sum of two tori including classes of examples for which no other method is known. We also include the application of this algebraic method to maps on the Klein bottle . Nielsen numbers for maps on were first calculated (geometrically) by Halpern. We include a sketch of Halpern's never published proof that for all maps on .

     

Kneading sequences for unimodal expanding maps of the interval

LetP andAC be two primary sequences with min{P, AC}≥RLR ^∞,ρ(P) and ρ(AC) be the eigenvalues ofP andAC, respectively. Letf∈C ⁰ (I, I) be a unimodal expanding map with expanding constant λ and m be a nonegative integer. It is proved thatf has the kneading sequenceK(f)≥(RC) ^*m *P if λ≥(ρ(P))^1/2m, andK(f)>(RC) ^*m*AC*E for any shift maximal sequenceE if λ>(ρ(AC))^1/2m. The value of (ρ(P))^1/2m or (ρ(AC))^1/2m is the best possible in the sense that the related conclusion may not be true if it is replaced by any smaller one. Project supported by the National Natural Science Foundation of China     

Invariant curves for birational surface maps

We classify invariant curves for birational surface maps that are expanding on cohomology. When the expansion is exponential, the arithmetic genus of an invariant curve is at most one. This implies severe constraints on both the type and number of irreducible components of the curve. In the case of an invariant curve with genus equal to one, we show that there is an associated invariant meromorphic two-form.

     

Competitive symbolic integration in satellite dynamics

In this work, we shall compare a symbolic integration with a numerical integration of a test problem arising in satellite dynamics. An equatorial satellite with the J₂ effect in BF variables will be taken. As symbolic integrator, we shall take erewton©, a Maple V package implemented by the authors. Classical and contrasted numerical methods will be chosen to perform the numerical integration. We pointed out the advantages of symbolic integration as an efficient method for making very long time predictions on the orbit of the satellite.     

Symbolic dynamics and Arnold diffusion

We consider hyperbolic tori of three degrees of freedom initially hyperbolic Hamiltonian systems. We prove that if the stable and unstable manifold of a hyperbolic torus intersect transversaly, then there exists a hyperbolic invariant set near a homoclinic orbit on which the dynamics is conjugated to a Bernoulli shift. The proof is based on a new geometrico-dynamical feature of partially hyperbolic systems, the transversality-torsion phenomenon, which produces complete hyperbolicity from partial hyperbolicity. We deduce the existence of infinitely many hyperbolic periodic orbits near the given torus. The relevance of these results for the instability of near-integrable Hamiltonian systems is then discussed. For a given transition chain, we construct chain of hyperbolic periodic orbits. Then we easily prove the existence of periodic orbits of arbitrarily high period close to such chain using standard results on hyperbolic sets.     

Nonsmooth one-dimensional maps: some basic concepts and definitions

The main purpose of the present survey is to contribute to the theory of dynamical systems defined by one-dimensional piecewise monotone maps. We recall some definitions known from the theory of smooth maps, which are applicable to piecewise smooth ones, and discuss the notions specific for the considered class of maps. To keep the presentation clear for the researchers working in other fields, especially in applications, many examples are provided. We focus mainly on the notions and concepts which are used for the investigation of various kinds of attractors of a map and related bifurcation structures observed in its parameter space.     

Resonance spectrum for one-dimensional layered media

We consider the “weighted” operator P_k=????_x a(x)? _x on the real line with a step-like coefficient which appears when propagation of waves through a finite slab of a periodic medium is studied. The medium is transparent at certain resonant frequencies which are related to the complex resonance spectrum of P_k. If the coefficient is periodic on a finite interval (locally periodic) with k identical cells, then the resonance spectrum of P_k has band structure. In the article, we study a transition to semi-infinite medium by taking the limit k→?∞?. The bands of resonances in the complex lower half plane are localized below the band spectrum of the corresponding periodic problem (k=∞) with k???1 or k resonances in each band. We prove that as k→?∞?, the resonance spectrum converges to the real axis.     

An effective hybrid algorithm for computing symbolic determinants

A hybrid algorithm for computing the determinant of a matrix whose entries are polynomials is presented. It is based on the dimension-decreasing algorithm [22] and the parallel algorithm for computing a symbolic determinant of [19]. First, through the dimension-decreasing algorithm, a given multivariate matrix can be converted to a bivariate matrix. Then, the parallel algorithm can be applied to effectively compute the determinant of the bivariate matrix. Experimental results show that the new algorithm can not only reduce enormously the intermediate expression swell in the process of symbolic computation, but also achieve higher degree of parallelism, compared with the single parallel algorithm given in [19].     

Experiments with a symbolic programming system for complex analysis

The paper reports the results of work on the construction of LISP programs for various symbolic operations in complex analysis, including the evaluation of integrals around closed contours by the use of Cauchy's Theorem. It is concluded that the only difficulty in the way of the preparation of programs for all of the important and useful textbook calculations in complex analysis is that large amounts of fast storage in the computer are needed.     

On co-σ-porosity of the parameters with dense critical orbits for skew tent maps and matching on generalised β-transformations

We prove that the critical point and the point 1 have dense orbits for Lebesgue-a.e., parameter pairs in the two-parameter skew tent family and generalised $\beta$-transformations. As an application, we show that for the generalised $\beta$-transformation with the tribonacci number as slope, there is matching (i.e., $T^n\left(0\right)=T^n\left(1\right)$ for some $n\ge 1$) for Lebesgue-a.e. translation parameter.