线段映射的局部变差增长与局部拓扑熵

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．

On Pointwise Variation Growth and Entropy of Interval Maps

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 Var (fn) 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 (fn) of the total variations Var (fn) 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° or on I° minus a fixed point. This is similar to the almost everywhere constancy of local entropy considered by Brin and Katok.