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 …     

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.     

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.     

单峰映射允许揉搓序列的维数和测度

本文证明了单峰映射的允许揉搓序列组成的集合在符号空间∑_2中的 Hausdorff 维数为1,1维 Hausdorff 测度为零。     

Computation of symbolic dynamics for one-dimensional maps

In this paper we design and implement rigorous algorithms for computing symbolic dynamics for piecewise-monotone-continuous maps of the interval. The algorithms are based on computing forwards and backwards approximations of the boundary, discontinuity and critical points. We explain how to handle the discontinuities in the symbolic dynamics which occur when the computed partition element boundaries are not disjoint. The method is applied to compute the symbolic dynamics and entropy bounds for the return map of the singular limit of a switching system with hysteresis and the forced Van der Pol equation.     

Minimal Cantor Systems and Unimodal Maps

We consider the restriction of unimodal maps f to the omega-limit set y ( c ) of the critical point for certain cases where y ( c ) is a Minimal Cantor set. We investigate the relation of these minimal systems to enumeration scales (generalized adding machines), to Vershik adic transformations on ordered Bratelli diagrams and to substitution shifts. Sufficient conditions are given for ( y ( c ), f ) to be uniquely ergodic.     

Local boundary dilatation of quasiconformal maps

By studying the existence problem of locally extremal Beltrami differential posed by F. Gardiner and N. Lakic, we introduce a new definition of the local boundary dilatations of points in Teichmuller spaces of simply connected plane domains, which is the same as the usual ones in the case of Jordan domains. Then we show that the answer to the problem of F. Gardiner and N. Lakic is affirmative according to this new definition. Comparing with the usual definitions, the problem of F. Gardiner and N. Lakic is partly answered and the results of G. Cui and Y. Qi is generalized.     

The number of rooted circuit boundary maps

In this article, the authors discuss two kinds of new planar maps： pan-fan maps and circuit boundary maps, and provide explicit expressions about their enumerating functions with different parameters. Meanwhile, two explicit counting formulas for circuit cubic boundary maps with two parameters; the size and the valency of the root-face, are also extracted.     

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     

Renormalization Operators and Kneading Sequences

ＲｅｎｏｒｍａｌｉｚａｔｉｏｎＯｐｅｒａｔｏｒｓａｎｄＫｎｅａｄｉｎｇＳｅｑｕｅｎｃｅｓ￥ＬｉａｏＧｏｎｇｆｕ（廖公夫）ａｎｄＺｈａｎｇＡｉｈｕａ（张爱华）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ．ＪｉｉｎＵｎｉｖｅｒｓｉｔｙ，Ｃｈａｎｇｃｈ...     

On the error-sum function of tent map base series

We first introduce tent map base series. The tent map base series is special case of generalized Lüroth series which has the tent map as a base map. Then we study some elementary properties of its error-sum function, and show that the function is continuous.     

On cascades of elliptic periodic points in two-dimensional symplectic maps with homoclinic tangencies

We study bifurcations of two-dimensional symplectic maps with quadratic homoclinic tangencies and prove results on the existence of cascade of elliptic periodic points for one and two parameter general unfoldings.      

Partial regularity results up to the boundary for harmonic maps into a Finsler manifold

We study the energy functional for maps from a Riemannian m-manifold M   into a Finsler space N=(Rn,F)$N=({\mathbb{R}}^n,F)$. Under the restriction 2?m?4$2?m?4$, we prove the full Hölder regularity of weakly harmonic maps (i.e., weak solutions of its Euler–Lagrange equation) from M to N   in the case that the Finsler structure F(u,X)$F(u,X)$ depends only on vectors X, and a partial Hölder regularity of energy minimizing maps in general cases.     

Conformal and semi-conformal biharmonic maps

We show that a conformal mapping between Riemannian manifolds of the same dimension n ≥ 3 is biharmonic if and only if the gradient of its dilation satisfies a certain second-order elliptic partial differential equation. On an Einstein manifold solutions can be generated from isoparametric functions. We characterise those semi-conformal submersions that are biharmonic in terms of their dilation and the fibre mean curvature vector field.      

具有稠密临界轨道和预不动临界轨道的单峰映射的丰富性

证明了对于实二次族在参数空间存在正Ｌｅｂｅｓｇｕｅ测度集合Ｅ,使得Ｅ中几乎所有的参数,相应的映射在不变测度的支集上具有稠密的临界轨道;还证明了Ｅ中存在稠密集合使得相应映射的临界轨道进入它的反向不动点。     

Geometric realization of Adams maps

Let G/P be a homogenous space with G a compact connected Lie group and P a connected subgroup of G of equal rank. As the rational cohomology ring of G/P is concentrated in even dimensions, for an integer k we can define the Adams map of type k to be l _k : H*(G/P, ℚ) → H*(G/P, ℚ), l _k (u) = k ⁱ u, u ∈ H ²ⁱ (G/P, ℚ). We show that if k is prime to the order of the Weyl group of G, then l _k can be induced by a self map of G/P. We also obtain results which imply the condition that k is prime to the order of the Weyl group of G is necessary.     

Transformations and non-degenerate maps

We shall prove the equivalences of a non-degenerate circle-preserving map and a Mobius transformation in Rn, of a non-degenerate geodesic-preserving map and an isometry in Hn, of a non-degenerate line-preserving map and an affine transformation in Rn. That a map is non-degenerate means that the image of the whole space under the map is not a circle, or geodesic or line respectively. These results hold without either injective or surjective, or even continuous assumptions, which are new and of a fundamental nature in geometry.     

On single-valuedness of convex set-valued maps

It is proved that ifX andY are linear spaces andF :X p(Y) is a set-valued map with convex graph such thatF(x) Ø for allx X andF(x ₀) is a singleton for somex ₀, thenF is single-valued and affine. Applications to metric projections and to adjoints of set-valued maps are given.Supported by NSF Grant DMS-9100228.The main result of this paper has been obtained while the second author was visiting the Pennsylvania State University in the framework of the exchange agreement between the Romanian Academy and the National Academy of Sciences of the U.S.A.     

Realization of numbers as the degrees of maps between manifolds

We determine the set of degrees between some classes of oriented closed （n - 1）-connected 2n-manifolds by using the arithmetic theory of quadratic forms.