Topological entropy ($h_{\mathrm{top}}$)

Topological entropy measures the exponential growth rate of distinguishable orbit segments of a dynamical system.

For a two-dimensional hyperbolic toral automorphism, the stable and unstable eigenvalues obey $|{\lambda }_s|<1<|{\lambda }_u|$. In a sufficiently small Bowen ball, lifting to the stable and unstable coordinates gives widths proportional to $ϵ$ and $ϵ|{\lambda }_u{|}^{-(n-1)}$. Its normalized Haar measure is therefore proportional to ${ϵ}^2|{\lambda }_u{|}^{-(n-1)}$, uniformly in the centre. Maximal separated sets cover by radius-$ϵ$ Bowen balls and pack disjoint radius-$ϵ/2$ Bowen balls, so their exponential growth rate is $\log |{\lambda }_u|$. Smallness of $ϵ$ must exclude nonzero lattice jumps between successive lifted iterates; this is essential to the ball-volume argument.

For a continuous map on a compact metric space, the Bowen metric compares the first $n$ points of two forward orbits. Its radius-$ϵ$ ball is $B_n(x,ϵ)=\{y:d_n(x,y)<ϵ\}$. If $s_n(ϵ)$ is the largest cardinality of an $ϵ$-separated set in this metric, then topological entropy is ${\lim }_{ϵ\downarrow 0}{\mathrm{limsup}}_{n\to \infty }{n}^{-1}\log s_n(ϵ)$. The uniform continuity of the identity between two compatible metrics gives the comparison of separated-set counts needed for independence of the original metric.

A radius-$ϵ$ ball for a Bowen metric consists of the points whose first $n$ iterates stay within $ϵ$ of the corresponding iterates of the centre. Balls at a fixed small radius distinguish forward orbit segments at that observational resolution.

A dynamical system has positive topological entropy when $h_{\mathrm{top}}>0$; 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.