Jun LUO(1), Yu HUANG(1), Zuo L
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.
