# 44. The Shadowing Theorem

Let $f \colon X \to X$ be a reversible dynamical system on a compact metric space. A *$\delta$-chain* is a sequence $(x_k)$, $k \in \mathbb{Z}$ such that

$$

d\big(f(x_k), x_{k+1}\big) \le \delta, \qquad \forall , k \in \mathbb{Z}.

$$

A point $y \in X$ is said to *$ \varepsilon$-shadow* a the $ \delta$-chain $(x_k)$ if

$$

d\big(f^k(y), x_k\big) \le \varepsilon, \qquad \forall , k \in \mathbb{Z}.

$$

Today we prove the *Shadowing Theorem*, which roughly speaking asserts that in the presence of hyperbolicity, for any $ \varepsilon>0$ there exists $ \delta >0$ such that every $ \delta$-chain is $ \varepsilon$-shadowed by exactly one point.

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