KolmogorovSinai

The Kolmogorov-Sinai entropy, often abbreviated as KS entropy or metric entropy, is a fundamental concept in dynamical systems theory. It quantifies the rate at which information is lost or, equivalently, the rate at which unpredictability increases in a dynamical system. It was independently developed by Andrey Kolmogorov and Yakov Sinai in the early 1960s.

The KS entropy is defined for ergodic dynamical systems. For a discrete-time dynamical system, it is the supremum of the average rate of information gain per unit time. Informally, it measures how quickly nearby trajectories diverge. A positive KS entropy indicates sensitive dependence on initial conditions, a hallmark of chaotic behavior. Systems with zero KS entropy are generally considered more predictable, though they may still exhibit complex behavior.

The calculation of KS entropy typically involves partitioning the state space into small cells and tracking how the number of these cells grows as the system evolves. The rate of this growth, when appropriately normalized and taken to the limit of infinitely small cells, yields the KS entropy. It is an invariant under topological conjugacy, meaning that two conjugate systems share the same KS entropy. This makes it a powerful tool for classifying and understanding the complexity of different dynamical systems, from simple iterated maps to complex physical phenomena.