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There are also notions of Markov and Lagrange spectra, which are related to height (instead of length) of geodesics in the modular surface. However, I am not sure that the vocabulary of spectrum for these predates the length spectrum.
Basically, the graph of this function starts from $(0,0)$, goes down to $(1/5, -1)$, goes up to $(4,/5, 2)$, goes down to $(6/5, 0)$, goes up to $(9/5, 3)$, etc. I will post a graph tomorrow to make the construction more intuitive (because it really is) ; it's a situation where a picture is worth a thousand words.
@Matt F.: No. If $x < 1/5$, then $T(x) = 1-5x$. Going back to the real number, we get $S(x) =1-5\{x\}+ \lfloor x \rfloor + F(\{x\}) = \lfloor x \rfloor + 1-5\{x\}-1 = \lfloor x \rfloor -5\{x\}$ whenever $\{x\} < 1/5$, so the sequence $(x_n)$ can decrease.
Given a Radon measure $\mu$ on $G \backslash X$, what if we just define $\psi(f) := \int_{G \backslash X} \sum_{x \in Gy} f(x) \ \text{d} \mu(y)$? This should define a $G$-equivariant positive linear functional on $\mathcal{C}_c (G)$, from which we conclude by the Riesz-Markov-Kakutani representation theorem. Is something missing?
No, it doesn't (e.g. you can have mutually singular invariant measures with full support). You basically get uniform convergence only in the case where the system has a unique invariant probability measure (unique ergodicity), but that's a pretty specific property.
@alesia: unfortunately, I don't know of one. I learnt it via study groups in ergodic theory (using Oseledets theorem, etc.), so without the vocabulary of Markov processes.
Just an idea: take $g(x) = 2x$. For any continuous $f$, the variation of $f$ on a small dyadic interval $[2^{-n}, 2^{-m}]$ goes to $0$ as $n$, $m \to + \infty$. If $f$ is random, then this variation converges almost surely, and thus in probability, to $0$. If the distribution of $f$ is $g$-invariant, then the variation of $f$ on any dyadic interval is $0$ almost surely, to $f$ is actually constant.
