Topological entropy measures the exponential growth rate of distinguishable orbit segments of a dynamical system.
A dynamical system has positive topological entropy when ; this expresses exponential orbit complexity.
A continuous interval map has positive topological entropy if and only if some positive iterate has a horseshoe for an interval map.
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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 \( X \).