作　　者： ; ; ;

机构地区： 中山大学数学与计算科学学院

出　　处： 《数学学报（中文版）》 2006年第2期311-316,共6页

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

领　　域： [理学] [理学]