具有不连续熵函数的移位映射
A shift map with a discontinuous entropy function
摘要：
设$f:X\to X$是紧致度量空间上具有有限拓扑熵的连续映射.此外，我们假设熵映射$μ\mapsto h_μ(f)$是上半连续的。众所周知，这意味着给定连续势$ϕ:X\to R$的定域熵函数的连续性。在本文中，我们证明了这一结果并不适用于高维势$Φ:X\to R^m$的情形。也就是说，我们构造了一个具有不连续局部熵函数的$f$维Lipschitz连续势$Φ$。
英文摘要：
Let $f:X\to X$ be a continuous map on a compact metric space with finite topological entropy. Further, we assume that the entropy map $μ\mapsto h_μ(f)$ is upper semi-continuous. It is well-known that this implies the continuity of the localized entropy function of a given continuous potential $ϕ:X\to R$. In this note we show that this result does not carry over to the case of higher-dimensional potentials $Φ:X\to R^m$. Namely, we construct for a shift map $f$ a $2$-dimensional Lipschitz continuous potential $Φ$ with a discontinuous localized entropy function.
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