Supercyclic and Hypercyclic Generalized Weighted Backward Shifts over a Non-Archimedean c0(N) Space

In the present paper, we propose to study generalized weighted backward shifts BB over non-Archimedean c0(N) spaces; here, B=(bij) is an upper triangular matrix with supi,j|bij|<∞. We investigate the sypercyclic and hypercyclic properties of BB. Furthermore, certain properties of the operator I+BB are studied as well. To establish the hypercyclic property of I+BB we have essentially used the non-Archimedeanity of the norm which leads to the difference between the real case.

KCC Analysis of a One-Dimensional System During Catastrophic Shift of the Hill Function: Douglas Tensor in the Nonequilibrium Region

This paper considers the stability of a one-dimensional system during a catastrophic shift described by the Hill function. Because the shifting process goes through a nonequilibrium region, we applied the theory of Kosambi, Cartan, and Chern (KCC) to analyze the stability of this region based on the differential geometrical invariants of the system. Our results show that the Douglas tensor, one of the invariants in the KCC theory, affects the robustness of the trajectory during a catastrophic shift. In this analysis, the forward and backward shifts can have different Jacobi stability structures in the nonequilibrium region. Moreover, the bifurcation curve of the catastrophic shift can be interpreted geometrically, as the solution curve where the nonlinear connection and the deviation curvature become zero. KCC analysis also shows that even if the catastrophic pattern itself is similar, the stability structure in the nonequilibrium region is different in some cases, from the viewpoint of the Douglas tensor.

Jℱ-Class Weighted Backward Shifts

In this article [Formula: see text]-class operators are introduced and some basic properties of [Formula: see text]-vectors are given. The [Formula: see text]-class operators include the [Formula: see text]-class operators and [Formula: see text]-class operators introduced by Costakis and Manoussos in 2008. This class also includes the [Formula: see text]-class and [Formula: see text]-class operators defined by Zhang [2012]. Furthermore, for the unilateral weighted backward shifts on a Fréchet sequence space, we establish a criterion under which the shift operators belong to the [Formula: see text]-class. From the criterion it is easy to obtain the existing criteria of hypercyclic backward shifts and of the topological mixing backward shifts. The obtained criterion also reveals the characteristic of [Formula: see text]-class shift operators by the recurrence property. Meanwhile, we obtain infinite topological entropy when the shifts have stronger recurrence property, which generalizes the related results by Brian et al. in 2017.