Eventually Periodic

An orbit $\{x_1,x_2,\dots \}$ is eventually periodic if it lands exactly on a periodic orbit after finitely many steps and cycles from there. Formally, $\exists$ integers $N\ge 0$, $k\ge 1$ such that:

$$f^{N+k}(x_1)=f^N(x_1)$$

Every EP orbit is also Asymptotically Periodic, landing on the cycle is a stronger condition than converging to it. Not every AP orbit is EP.

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The canonical example: under $f(x)=3x(\mathrm{mod}1)$, all rationals are EP. The map acts as a left-shift on base-3 digits, so any eventually-repeating base-3 expansion hits a cycle in finite steps (proven in HW2). e.g.

$$\frac{1}{12}\mapsto \frac{1}{4}\mapsto \frac{3}{4}\mapsto \frac{1}{4}\mapsto \dots$$

one transient step, then locked into the period-2 source $\left\{\frac{1}{4},\frac{3}{4}\right\}$.

The Lyapunov exponent of an EP orbit equals that of the target cycle — the transient washes out in the limit.