I would appreciate a reference to support this statement that
appears under the **Geodesic** entry of the
[CRC Encyclopedia of Mathematics](https://www.routledge.com/The-CRC-Encyclopedia-of-Mathematics-Third-Edition---3-Volume-Set/):



> 
> "no matter how badly a sphere is distorted,
> there exists an infinite number of closed geodesics on it.
> This general result, demonstrated in the early 1990's,
> extended earlier work of Birkhoff, ..."
> 
> 
> 



Infinite number of closed geodesics on distorted sphere
Will Jagy answered my question:



> 
> Bangert, Victor. "On the existence of closed geodesics on two-spheres." *International Journal of Mathematics* 4, no. 01 (1993): 1-10.
> [doi](https://doi.org/10.1142/S0129167X93000029).
> 
> 
> 


"...one obtains the existence of infinitely many closed geodesics for every Riemannian metric on $\mathbb{S}^2$."


