[Fekete's (subadditive) lemma](http://en.wikipedia.org/wiki/Subadditivity#Properties) 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](http://en.wikipedia.org/wiki/Banach_algebra#Spectral_theory) 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.



Generalizations and relative applications of Fekete's subadditive lemma
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.


References:


* Schreiber. J. Diff. Eq. 148 (1998), 334--350.
* Sturman, Stark. Nonlinearity 13 (2000), 113--143.
* Morris. Proc. London Math. Soc. (3) 107 (2013) 121–150. See Appendix A.


