线段映射的局部变差增长与局部拓扑熵  

On Pointwise Variation Growth and Entropy of Interval Maps

作　　者：黄煜[1] 罗俊[1] 周作领[2] 

机构地区：[1]中山大学数学与计算科学学院,广州510275 [2]中山大学岭南学院,广州510275

出　　处：《数学学报（中文版）》2006年第2期311-316,共6页Acta Mathematica Sinica：Chinese Series

基　　金：国家自然科学基金项目;广东省自然科学基金资助项目;中山大学高等学术中心基金资助项目

摘　　要：本文考虑闭区间上变差有界的连续映射f:I→I的局部变差增长γ(x,f)与局部拓扑熵h(x,f)．将证明γ(x,f)≥h(x,f)对所有x∈I成立,并且局部变差增长映射γf(x)=γ(x,f)与局部拓扑熵映射sf(x)=h(x,f)都是上半连续的,得到一个变分原理:局部变差增长γ(x,f)与局部拓扑熵h(x,f)的上确界分别等于全局变差增长γ(f)=limn→∞1/nln Var(fn)与拓扑熵h(f)．当映射f:I→I拓扑传递时,与Brin 和Katok对局部(测度)熵的讨论类似,我们证明,至多除一个不动点外,局部变差增长γ(x,f)与局部拓扑熵h(x,f)在开区间I°内恒为常值．This note considers the local growth rate of variation γ（x, f） and local topological entropy h（x, f） at points x ∈ I for a continuous map f on a compact interval I such that the total variation Vat （f^n） is bounded for all n ≥ 0. We will show that γ（x, f） is always no less than h（x, f） and that the functions x →γ （x, f） and x → h（x, f）, which map a point x to its local growth rate of variation and its local topological entropy respectively, are both upper semi-continuous. We also obtain a variational principle： the supremum of the local growth rate of variation and that of local topological entropies are equal to the global exponential growth rate γ（f） = limn→∞1/n ln Var（f^n） of the total variations Var （f^n） and the topological entropy h（f）, respectively. When the map f ： I → I is topologically transitive, we infer that the local growth rates of variation and local topological entropies functions are both constant on I^o or on I^o minus a fixed point. This is similar to the almost everywhere constancy of local entropy considered by Brin and Katok.

关 键 词：有界变差 变分原理 熵 

分 类 号：O182.1[理学—数学]