Algebraic dichotomies with an application to the stability of Riemann solutions of conservation laws

A Hartman-Grobman theorem for algebraic dichotomies

“…In the present paper, we try to reduce the second condition. Motivated by Lin's algebraic dichotomy (see Lin [1]) and the works of Palmer [22] and Zhang et al [38], we study the C 0 linearization with the algebraic dichotomy. Moreover, we prove that both the homeomorphism H and its inverse G = H −1 are Hölder continuous.…”

“…Fenner and Pinto [9]), nonuniform (µ, ν)-dichotomy (see Bento and Silva [10], Chang et al [11], Barreira et al [12], Chu [13]), algebraic dichotomy (see Lin [1]). In this paper, we pay particular attention to the algebraic dichotomy introduced by Lin [1]. Lin discussed the basic properties of the algebraic dichotomy, and calculated the power of the weight function.…”

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Algebraic dichotomy is a generalization of an exponential dichotomy (see Lin [1]). This paper gives a version of Hartman-Grobman linearization theorem assuming that linear system admits an algebraic dichotomy, which generalizes the Palmer's linearization theorem.

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On (h,k)-Dichotomies for Nonautonomous Linear Difference Equations in Banach Spaces

“…Indeed, assume that the system (A) admits a (ℎ, )dichotomy. Then, by (14) and 17, we get the inequalities…”

“…One important and useful concept of dichotomy in the study of difference equations is the so-called (uniform) (ℎ, )dichotomy concept introduced by Pinto in [10]. Since then, this concept has been extensively studied and applied; see, for example, Naulin and Pinto [11], Megan [12], Fenner [13], and Lin [14] where several examples are presented.…”