Let $n$ and $p$ be two integers greater than or equal to 2. We fix a sequence $\left( A _ { k } \right) _ { k \in \mathbb { N } }$ of matrices of $\mathrm { GL } _ { n } ( \mathbb { C } )$ which is $p$-periodic. The sequence $\left( \Phi _ { k } \right) _ { k \in \mathbb { N } }$ is defined by $\Phi _ { 0 } = I _ { n }$ and $\Phi _ { k + 1 } = A _ { k } \Phi _ { k }$. We denote by $B$ a matrix in $\mathrm { GL } _ { n } ( \mathbb { C } )$ satisfying $B ^ { p } = \Phi _ { p }$, and $\left( P _ { k } \right) _ { k \in \mathbb { N } }$ the unique $p$-periodic sequence such that $\Phi _ { k } = P _ { k } B ^ { k }$. Let $\left( Y _ { k } \right) _ { k \in \mathbb { N } }$ be a solution of (III.1). a) Prove that if $\lim _ { k \rightarrow + \infty } \left\| B ^ { k } \right\| _ { 0 } = 0$, then $\lim _ { k \rightarrow + \infty } \left\| Y _ { k } \right\| _ { \infty } = 0$. b) Prove that if the sequence $\left( \left\| B ^ { k } \right\| _ { 0 } \right) _ { k \in \mathbb { N } }$ is bounded, then the sequence $\left( \left\| Y _ { k } \right\| _ { \infty } \right) _ { k \in \mathbb { N } }$ is also bounded.
Let $n$ and $p$ be two integers greater than or equal to 2. We fix a sequence $\left( A _ { k } \right) _ { k \in \mathbb { N } }$ of matrices of $\mathrm { GL } _ { n } ( \mathbb { C } )$ which is $p$-periodic. The sequence $\left( \Phi _ { k } \right) _ { k \in \mathbb { N } }$ is defined by $\Phi _ { 0 } = I _ { n }$ and $\Phi _ { k + 1 } = A _ { k } \Phi _ { k }$. We denote by $B$ a matrix in $\mathrm { GL } _ { n } ( \mathbb { C } )$ satisfying $B ^ { p } = \Phi _ { p }$, and $\left( P _ { k } \right) _ { k \in \mathbb { N } }$ the unique $p$-periodic sequence such that $\Phi _ { k } = P _ { k } B ^ { k }$. Let $\left( Y _ { k } \right) _ { k \in \mathbb { N } }$ be a solution of (III.1).
a) Prove that if $\lim _ { k \rightarrow + \infty } \left\| B ^ { k } \right\| _ { 0 } = 0$, then $\lim _ { k \rightarrow + \infty } \left\| Y _ { k } \right\| _ { \infty } = 0$.
b) Prove that if the sequence $\left( \left\| B ^ { k } \right\| _ { 0 } \right) _ { k \in \mathbb { N } }$ is bounded, then the sequence $\left( \left\| Y _ { k } \right\| _ { \infty } \right) _ { k \in \mathbb { N } }$ is also bounded.