# Generalization(s) of Subadditive Ergodic Theorems

I am interested in dynamical gadgets which can be described by sampling along the orbits of points in some ergodic system $(\Omega,\mu,T)$. When $\mu$ is a probability measure, the theory of such objects is quite well understood, so I would like to study what happens when one allows the measure $\mu$ to be a properly $\sigma$ - finite measure (i.e. infinite and $\sigma$-finite). In this setting, a $\mu$-preserving transformation $T:\Omega \to \Omega$ is called ergodic if $T^{-1}E=E \Longrightarrow \mu(E) = 0$ or $\mu (\Omega - E) = 0$.

In this setting, some facts from finite ergodic theory carry over - for example, one still has that any $T$-invariant measurable function $\Omega \to \mathbb{R}$ must be $\mu$-almost surely constant. However, some of the nicer results, such as the Birkhoff ergodic theorem fail to be true.

In the probability measure setting, we have the following very pleasant subadditive ergodic theorem, due to Kingman:

Theorem (Kingman): Suppose $(\Omega,\mu,T)$ is ergodic, and $f_n:\Omega \to \mathbb{R}$ are measurable, obey the subadditivity condition $f_{n+m}(x) \leq f_n(x)+f_m(T^n(x))$, and satisfy $\|f_n\|_{\infty} \leq C\cdot n$. Then the limit

$\displaystyle\lim_{n \to \infty}\frac{1}{n}\int_{\Omega} f_n(x) d \mu(x)$

exists and is equal to $\lim_{n \to \infty}\frac{1}{n} f_n(x)$ for $\mu$ almost every $x \in \Omega$.

My question is the following: does Kingman carry over to the infinite measure setting? If not, are there any weakened generalizations that one can obtain if one makes additional "niceness" assumptions about $\mu$ and\or $T$?

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Birkhoff's theorem is a corollary of the subadditive ergodic theorem so you can't hope for a better subadditive theorem than an ergodic theorem – Anthony Quas Apr 22 '12 at 7:36

I have not ever tried to use it, but there is some infinite measure generalization of the subadditive ergodic theorem in the spirit of the ratio ergodic theorem in this paper:

Akcoglu, M. A.; Sucheston, L.

A ratio ergodic theorem for superadditive processes. Z. Wahrsch. Verw. Gebiete 44 (1978), no. 4, 269–278. 28D05 (60F15)

Here is the math review:

A Markov operator is a positive linear operator $T$ on $L_1(X,F,μ)$ such that $T*1=1$. A sequence of $L_1+$ functions $f_0$,$f_1$,$f_2$,⋯ is superadditive if $s_k+n \geq s_k+T^k s_n$, where $s_n=f_0+f_1+⋯+f_n−1$ and $s_0=0$. An exact dominant of such a sequence is an $L_1^+$ function $δ$ such that $\sum_{i=0}^{n-1} T^i δ≥ s_n$ and $\int δ \, dμ= \lim_n\frac{1}{n}\int s_n \, dμ$. The authors show that a Markov operator always has an exact dominant, by generalizing an earlier idea of J. F. C. Kingman [J. Roy. Statist. Soc. Ser. B 30 (1968), 499--510; MR0254907 (40 #8114)]. The authors then use their result to prove a generalization of Kingman's ergodic theorem and the R. V. Chacon and D. S. Ornstein theorem [Illinois J. Math. 4 (1960), 153--160; MR0110954 (22 #1822)].

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Maybe you can read K. Schurger. Almost subadditive extensions of Kingman's ergodic theorem. Ann. Probab. 1991. But not sure why you are interesting in it.

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