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By K. Mathiak, Steffen Supra

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6. Let g ∈ G be hyperbolic, with eigenvalue Λg corresponding to the attracting fixed point αg+ . Let Λ(g) := | log Λg |, and let be the period of the continued fraction expansion of β = αg+ . Then {{∗, β}}G = = 1 λ(β) k=1 {0, g(0)}G Λ(g) {gk−1 (β) · 0, gk−1 (β) · i∞}G . This shows that, in this case, the limiting modular symbols are linear combinations of classical modular symbols, with coefficients in the field generated over Q by the Lyapunov exponents λ(β) of the quadratic irrationalities. In terms of geodesics on the modular curve, this is the case where the geodesic has a limiting cycle given by the closed geodesic {0, g(0)}G (Figure 6).

71]). The weak convergence proved by this type of argument can be improved to strong convergence by applying the strong law of large numbers to the “random variables” ϕk = ϕ ◦ T k (cf. [71] for details). The result effectively plays the role of an ergodic theorem for the shift T on E. 5. Hecke eigenforms A very important question is what happens to modular forms at the noncommutative boundary of the modular curves. There is a variety of phenomena in the theory of modular forms that hint to the fact that a suitable class of “modular forms” survives on the noncommutative boundary.

Ergodicity) via the spectral theory of an associated operator. This allows one to employ techniques of functional analysis and derive conclusions on dynamics. 30) (Lσ f )(x, t) = k=1 1 f (x + k)2σ depending on a complex parameter σ. 1 , x+k 0 1 1 k ·t 36 2. 31) (Lσ f )(x, t) = exp(h(y, s)) f (y, s), (y,s)∈T −1 (x,t) where we take h(x, t) = −2σ log |T (x, t)|. Clearly this operator is well suited for capturing the dynamical properties of the map T as it is defined as a weighted sum over preimages.

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