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.

Here is the proof that an orientation preserving homeomorphism $f$ of $\mathbb T$ has a well-defined rotation number. Let $F:{\mathbb R}\rightarrow\mathbb R$ be its lift. It is an increasing function verifying $F(x+\ell)=F(x)+\ell$ for every integer $\ell$. Let $x\in\mathbb R$ be given and $u_n=F^{(n)}(x)$. We have to prove that $\frac1nu_n$ has a finite limit. To do so, fix $n$ and define $N$ so that $u_n\in[x+N,x+N+1)$. Then $$u_{n+m}=F^{(m)}(u_n)\in[F^{(m)}(x+N),F^{(m)}(x+N+1))=[F^{(m)}(x)+N,F^{(m)}(x)+N+1).$$ This gives $u_{n+m}\in[u_m+N,u_m+N+1)$. Consequently, we obtain $$u_m+u_n-1-x\le u_{n+m}\le u_m+u_n+1-x.$$ Applying Fekete's Lemma to $v_n=u_n+1-x$, we see that $\frac1nv_n$, hence $\frac1nu_n$, has a limit $\rho<+\infty$. Applying it to $w_n=u_n-1-x$, we see that this limit is finite.

Finally, the limit does not depend upon the starting point $x$, because if $x\le y\le x+1$, then $F^{(m)}(x)\le F^{(m)}(y)\le F^{(m)}(x)+1$. An other use of the monotonicity shows that the limit is the same as $n\rightarrow-\infty$.

This lemma is simple, but it is very useful in rigorous proofs of Statistical Mechanics.

For example the existence of the thermondynamic limit of the free energy per particule $\frac{f_N}{N}$ of an Ising spin model can be proven in many cases by proving the free energy $f_N$ is sub/super-additive. If interested you can see:

F. Guerra, F. Toninelli - The Thermodynamic Limit in Mean Field Spin Glass Models .

If someone know any generalisation of this lemma, I'm very interested too.