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I would appreciate a reference to support this statement that appears under the Geodesic entry of the CRC Encyclopedia of Mathematics:

"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, ..."

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    $\begingroup$ Bangert, 1993, worldscientific.com/doi/abs/10.1142/… Refers to J. Franks, 1992, Inventiones $\endgroup$
    – Will Jagy
    Commented Jan 16, 2022 at 2:07
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    $\begingroup$ good survey arxiv.org/pdf/1308.5417.pdf $\endgroup$
    – Will Jagy
    Commented Jan 16, 2022 at 2:16
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    $\begingroup$ @MoisheKohan - On page 27 we find "Theorem 6.14 (Main Result). Every closed Riemannian manifold M admits infinitely many geometrically distinct, non-constant, prime closed geodesics." There is no hypothesis on the dimension of M (to rule out the point and the circle). Trying to understand the first sentence of the proof (the definition of X given before 6.5) left me no wiser. $\endgroup$
    – Sam Nead
    Commented Jan 16, 2022 at 10:25
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    $\begingroup$ @SamNead: Maybe I will post a question about the current status of the (notorious) conjecture on closed geodesics. The above reference claims a solution. By now the paper is 4 year old. Of course, especially during the pandemic, refereeing may take long time... $\endgroup$
    – Moishe Kohan
    Commented Jan 16, 2022 at 22:41
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    $\begingroup$ mathoverflow.net/questions/152513/… $\endgroup$
    – Ali Taghavi
    Commented Jan 17, 2022 at 12:56

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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.

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

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