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Fabian Haiden studies connections among geometry, dynamical systems and category theory. His course notes Dynamics, Stability, and Noncommutative Algebra by Fabian Haiden develop topological entropy using covers and separated or spanning sets of trajectories, making orbit complexity into a quantitative invariant.

His research page connects these themes with stability conditions and categorical constructions. This illustrates how different mathematical descriptions can carry information about the same dynamics.^[ref]

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.