Fekete's (subadditive) lemma takes its name from a 1923 paper by the Hungarian mathematician Michael Fekete [1]. A historical overview and references to (a couple of) generalizations and applications of the result are found in Steele's book on probability and combinatorial optimization [2, Section 1.10], where a special mention is made to the work of Pólya and Szegő on the structure of real sequences and series [3, Ch. 3, Sect. 1] and that of Hammersley [4], motivated by percolation theory, on subadditive functions, the
continuous analogue of subadditive sequences, whose systematic study was initiated, as far as I know, by Hille and Phillips in the 1957 edition of their beautiful monograph on functional analysis and semigroups [5, Ch. VII]. The same Steele acknowledges that his own 1989 proof of Kingman's subadditive ergodic theorem [6], of which Birkoff's celebrated theorem is a corollary, was eventually inspired by Fekete's lemma. Now, my question is:
Can you point out further generalizations (and corresponding
(interesting) applications) of Fekete's lemma?
Added later. Fekete's lemma can be used to prove that the limit occurring in the spectral radius formula does actually exist. And this counts (to me) as an (interesting) application.
Bibliography.
[1] M. Fekete (1923), Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit. ganzzahligen Koeffizienten, Math. Zeit., Vol. 17, pp. 228-249.
[2] M.J. Steele, Probability theory and combinatorial optimization, SIAM, Philadelphia, 1997.
[3] G. Pólya and G. Szegő, Problems and Theorems in Analysis, Vol. I, Springer-Verlag, Berlin, 1998 (reprint of the 1978 Edition).
[4] J.M. Hammersley (1962), Generalization of the fundamental theorem of subadditive functions, Proc. Cambridge Philos. Soc.,
Vol. 58, pp. 235-238.
[5] E. Hille and R.S. Phillips, Functional analysis and
semi-groups, American Math. Soc., 1996 (revised edition).
[6] J.M. Steele (1989), Kingman's subadditive ergodic theorem, Annales de l'I.H.P., Section B, Vol. 25, No. 1, pp. 93-98.
Best Answer
Since you mentioned Kingman's subadditive ergodic theorem, you may find interesting the following semi-uniform subadditive ergodic theorem:
Let $T \colon X \to X$ be a continuous map of a compact metric space $X$. If $f_n \colon X \to [-\infty,+\infty)$ is a subadditive sequence ($f_{n+m} \le f_n + f_m \circ T^n$) of upper semicontinuous functions then: $$ \sup_{\mu} \lim_{n \to \infty} \frac{1}{n} \int_X f_n d\mu = \lim_{n \to \infty} \frac{1}{n} \sup_{x \in X} f_n(x) , $$ where the first $\sup$ is taken over all $T$-invariant probability measures.
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