 Topological entropy

Topological entropy measures the exponential growth of distinguishable orbit histories. For a continuous map $F$ on a compact metric space, compare two initial states by the largest distance between their first $n$ iterates:Let $s_n(ϵ)$ be the largest number of initial states whose histories are pairwise more than $ϵ$ apart in this metric. ThenThis is a count at fixed observational resolution before the resolution is refined.^[ref]

The identity map has zero topological entropy: observing longer does not distinguish additional histories at fixed resolution. In the binary symbolic dynamics of a full two-symbol shift, there are $2^n$ length-$n$ words, giving entropy $\log 2$. Thus positive topological entropy expresses persistent exponential orbit complexity, rather than merely a large instantaneous displacement.

Topological entropy is a concept in dynamical systems that provides a measure of the complexity of a system. It quantifies the rate at which information about the state of a dynamical system is lost over time, reflecting the system's unpredictability or chaotic behavior. More formally, topological entropy is defined for a continuous map \( f: X \to X \) on a compact metric space \( X \).