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 $\Phi _ { k + 1 } = A _ { k } \Phi _ { k }$. The matrix $\Phi _ { p }$ is called the Floquet matrix of equation (III.1) and its complex eigenvalues are called the Floquet multipliers of (III.1). Let $\rho$ be a Floquet multiplier of (III.1). a) Prove that there exists a nonzero solution $\left( Y _ { k } \right) _ { k \in \mathbb { N } }$ of (III.1) satisfying $\forall k \in \mathbb { N } , Y _ { k + p } = \rho Y _ { k }$. b) Let $\left( Y _ { k } \right) _ { k \in \mathbb { N } }$ be such a solution, prove that, if $| \rho | < 1 , \lim _ { k \rightarrow + \infty } \left\| Y _ { k } \right\| _ { \infty } = 0$.
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 $\Phi _ { k + 1 } = A _ { k } \Phi _ { k }$. The matrix $\Phi _ { p }$ is called the Floquet matrix of equation (III.1) and its complex eigenvalues are called the Floquet multipliers of (III.1). Let $\rho$ be a Floquet multiplier of (III.1).
a) Prove that there exists a nonzero solution $\left( Y _ { k } \right) _ { k \in \mathbb { N } }$ of (III.1) satisfying $\forall k \in \mathbb { N } , Y _ { k + p } = \rho Y _ { k }$.
b) Let $\left( Y _ { k } \right) _ { k \in \mathbb { N } }$ be such a solution, prove that, if $| \rho | < 1 , \lim _ { k \rightarrow + \infty } \left\| Y _ { k } \right\| _ { \infty } = 0$.