Periodic points of the doubling map and toral automorphisms

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Problem 1.3.12 from Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. "Problems and Solutions in Mathematics" [CJP+11]More problems in Section 1.3 "Ring Theory"

Define $f : [0,1) \to [0,1)$ by

f(x) = \begin{cases} 2x, & \text{if } 0 \leq 2x < 1, \\ 2x - 1, & \text{if } 1 \leq 2x < 2. \end{cases}

Find all $x$ such that

f(f(f(f(f(f(x)))))) = x.

Problem 2.1.18 from Steeb W., Hardy Y. "Problems and Solutions in Introductory and Advanced Matrix Calculus" [StHa17]More problems in Section 2.1 "Chapter Problems"

Consider the area-preserving map of the two-dimensional torus (modulo 1)

\begin{pmatrix} x_{1}' \\ x_{2}' \end{pmatrix} = A \begin{pmatrix} x_{1} \\ x_{2} \end{pmatrix}, \qquad A = \begin{pmatrix} 4 & 15 \\ 1 & 4 \end{pmatrix}

where $\operatorname {det}\left(A\right) = 1$ (area-preserving). Consider a rational point on the torus

\begin{pmatrix} x_{1} \\ x_{2} \end{pmatrix} = \begin{pmatrix} n_{1}/p \\ n_{2}/p \end{pmatrix}

where $p$ is a prime number (except 2, 3, 5) and $n_{1}$ , $n_{2}$ are integers between 0 and $p-1$ . One finds that the orbit has the following property. It is periodic and its period $T$ depends on $p$ alone. Consider $p = 7$ , $n_{1} = 2$ , $n_{2} = 3$ . Find the orbit and the period $T$ .

References

[CJP+11] Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. Problems and Solutions in Mathematics. 2nd ed., World Scientific, 2011.

[StHa17] Steeb W., Hardy Y. Problems and Solutions in Introductory and Advanced Matrix Calculus. 2nd ed., World Scientific Publishing, 2017.