Theorem 3

No orbit of the Gauss map has a Lyapunov exponent smaller than .

Proof

Let be any initial point in such that exists. We will show that the product which appears in the definition of must be at least (for N sufficiently large) which will prove the theorem. We consider two subsequent elements and of the orbit of . If k=N, enlarge the product by one term. Note and are related by . If then the contribution of to the product is at least . If instead then so the contribution of to the product is at least . This proves the theorem.