Continuity and Invariance of the Sacker–Sell Spectrum

A spectral dichotomy version of the nonautonomous Markus–Yamabe conjecture

“…This property can be easily proved by using Proposition 2. In addition, the above result is inspired in the work of Pötzsche and Russ [20,Prop.8], which studied the invariance of the spectrum in the discrete case. To the best of our knowledge, there are no results in the continuous case.…”

Some relations between Bohl exponents and the exponential dichotomy spectrum

“…As pointed out in [12], the Bohl's exponents β A (ξ) and β A (ξ) can be seen as measures of the biggest and smallest growth rate of the solution k → X(k, 0)ξ of (1). To the best of our knowledge, the Bohl's exponents for discrete systems have been studied firstly by Ben-Artzi and Gohberg [8] as a corresponding version of the continuous case studied in [11, Ch.III], they have also been defined in an alternative but equivalent formulation by Pötzsche in [18,19,20] for algebraic and scalar difference equations.…”

“…The spectral theory associated to the exponential dichotomy has been extensively developed for discrete and continuous systems both in finite [1,20] and infinite dimensions [21]. In order to contextualize our main result, We will recall a fundamental result:…”

“…The relation between Bohl's exponents and the exponential dichotomy spectrum of (1) has been deeply studied on the half line Z + 0 both in the continuous [12] and in the discrete case [19,20]. Nevertheless, there are less results when considering the whole line and this note can be seen as a contribution in this context.…”

The Bohl Spectrum for Linear Nonautonomous Differential Equations

“…Note that the Sacker-Sell spectrum is upper semi-continuous in general, and in [26], sufficient criteria for continuity of the Sacker-Sell spectrum are established.…”