Exercise 19 solution: First note eo(x_{0}, F) + 1 = { F^{i}(x_{0}) + j + 1 : i, j ∈ Z } = { F^{i}(x_{0}) + k : i, k ∈ Z } = eo(x_{0}, F). Also, if x = F^{i}(x_{0}) + j ∈ eo(x_{0}, F), then H(x + 1) = H(F^{i}(x_{0}) + j + 1) = iα + j + 1 = H(x) + 1.

Let y ∈ R. We have

H(y + 1) = sup\{ \overline{H}(x) : x ∈ eo(x_{0}), x < y + 1 \}
= sup\{ \overline{H}(x − 1) + 1 : x ∈ eo(x_{0}), x − 1 < y \}
= sup\{ \overline{H}(x) + 1 : x ∈ eo(x_{0}), x < y \}
= H(y) + 1.