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Exponential dichotomy and trichotomy for difference equations. (English) Zbl 0939.39003

The authors focus their attention on some properties of exponential dichotomy and trichotomy of the linear difference equation

$x\left(n+1\right)=A\left(n\right)x\left(n\right),\phantom{\rule{4pt}{0ex}}n\in ℤ,$

where $A\left(n\right)$ is a $d×d$ invertible matrix for each $n\in ℤ$. Under the assumptions that this equation has an exponential dichotomy or trichotomy and the $d×d$ invertible matrix $B\left(n\right)$ is such that $A\left(n\right)+B\left(n\right)$ is invertible they prove that the perturbed equation

$x\left(n+1\right)=\left(A\left(n\right)+B\left(n\right)\right)x\left(n\right),\phantom{\rule{4pt}{0ex}}n\in ℤ$

has an exponential dichotomy or trichotomy too, if the norm of $B\left(n\right)$ is sufficiently small. This result improves some known result on the invariance of exponential dichotomy and trichotomy under some perturbations because the radius of the perturbation considered in the paper is larger than those known. Besides the equivalence between the exponential dichotomy for linear difference equations with almost periodic coefficients in an infinite integer interval and in a finite sufficiently long integer interval is proved. This statement is a discrete version of the corresponding equivalence for an almost periodic differential equation $\frac{dx}{dt}=A\left(t\right)x$.

MSC:
 39A10 Additive difference equations 34D09 Dichotomy, trichotomy 34C27 Almost and pseudo-almost periodic solutions of ODE