Positive top Lyapunov exponent via invariant cones: single trajectories (Q465418)

Given \(\lambda\geq 1\), the authors consider the quadratic forms in \(\mathbb R^p\), \(p\geq 2\), of type \((k,p-k)\) with arbitrary integer \(k\in [1,p-1]\), defined by NEWLINE\[NEWLINEQ_\lambda(x_1,\dots,x_p)=x_1^2+\cdots+x_k^2-\lambda x_{k+1}^2-\cdots-\lambda x_p^2,NEWLINE\]NEWLINE and the corresponding cone NEWLINE\[NEWLINEC_\lambda=\left\{ x\in\mathbb R^p:Q_\lambda (x)>0\right\}\cup\{0\}.NEWLINE\]NEWLINE Also, they consider the families of matrices NEWLINE\[NEWLINE\mathcal{F}_\lambda=\left\{A\in\mathrm{GL}_p:\, Q_\lambda(Ax)>0 \text{ for } x\in \overline{C}_1\setminus \{0\}\right\}NEWLINE\]NEWLINE and NEWLINE\[NEWLINES\mathcal{F}_\lambda=\left\{ A\in\mathcal{F}_\lambda:\,|\det(A)|=1\right\}.NEWLINE\]NEWLINE The main results of the paper establish criteria for the positivity of the top Lyapunov exponent of a nonautonomous dynamics with discrete time in terms of the family of matrices \(S\mathcal{F}_\lambda\) for \(\lambda>1\). In fact, it is proved that if \((A_n)_{n\geq 1}\) is a sequence of matrices in \(S\mathcal{F}_\lambda\) for some \(\lambda>1\), then the top Lyapunov exponent of the corresponding cocycle is positive. Furthermore, the authors consider similar results in the case of flows, and expose the relation to ergodic theory.NEWLINENEWLINENEWLINEThe results in this paper extend, improve and unify some existing results in the field.