Let ([sum ]_A, T) be a topologically mixing subshift of finite type on an alphabet consisting of m symbols and let Φ:[sum ]_A → R^d be a continuous function. Denote by σ_Φ(x) the ergodic limit lim_n→∞n⁻¹ [sum ]ⁿ⁻¹_j=0 Φ(T^jx) when the limit exists. Possible ergodic limits are just mean values ∫ Φdμ for all T-invariant measures. For any possible ergodic limit α, the following variational formula is proved:[formula here]where h_μ denotes the entropy of μ and h_top denotes topological entropy. It is also proved that unless all points have the same ergodic limit, then the set of points whose ergodic limit does not exist has the same topological entropy as the whole space [sum ]_A

Assume that x∈[0,1) admits its continued fraction expansion x=[a₁(x),a₂(x),…]. The Khintchine exponent γ(x) of x is defined by $\gamma (x):=\lim _{n\to \infty }({1}/{n}) \sum _{j=1}^n \log a_j(x)$ when the limit exists. The Khintchine spectrum dim E_ξ is studied in detail, where E_ξ:={x∈[0,1):γ(x)=ξ}(ξ≥0) and dim denotes the Hausdorff dimension. In particular, we prove the remarkable fact that the Khintchine spectrum dim E_ξ, as a function of $\xi \in [0, +\infty )$ , is neither concave nor convex. This is a new phenomenon from the usual point of view of multifractal analysis. Fast Khintchine exponents defined by $\gamma ^{\varphi }(x):=\lim _{n\to \infty }({1}/({\varphi (n)}))\sum _{j=1}^n \log a_j(x)$ are also studied, where φ(n) tends to infinity faster than n does. Under some regular conditions on φ, it is proved that the fast Khintchine spectrum dim ({x∈[0,1]:γ^φ(x)=ξ}) is a constant function. Our method also works for other spectra such as the Lyapunov spectrum and the fast Lyapunov spectrum.

Bescovitch considered the Hausdorff dimensions of sets related to the sum of digits of real numbers represented in the dyadic expansions. In this paper, we provide an analogy for continued fraction expansions over the field of formal Laurent series. We also calculate the Hausdorff dimensions of sets of Laurent series which have given polynomial or exponential approximation orders. Such approximations converge faster than those of almost all Laurent series (with respect to the Haar measure).