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A sofic shift is a shift space consisting of bi-infinite labels of paths from a labelled graph. Being a dynamical system, the distribution of its closed orbits may indicate the complexity of the shift. For this purpose, prime orbit and Mertens’ orbit counting functions are introduced as a way to describe the growth of the closed orbits. The asymptotic behaviours of these counting functions can be implied from the analyticity of the Artin–Mazur zeta function of the shift. Its zeta function is expressed implicitly in terms of several signed subset matrices. In this paper, we will prove the asymptotic behaviours of the counting functions for sofic shifts via their zeta function. This involves investigating the properties of the said matrices. Suprisingly, the proof simply uses some well-known facts about sofic shifts, especially on the minimal right-resolving presentations. Furthermore, we will demonstrate this result by revisiting the case for periodic-finite-type shifts, which are a particular type of sofic shifts. At the end, we will briefly discuss the application of our finding towards the finite group and homogeneous extensions of a sofic shift. [ABSTRACT FROM AUTHOR]

The objective of this article is to introduce function weighted L-R-complete dislocated quasi-metric spaces and to present fixed point results fulfilling generalized rational type F-contraction for a multivalued mapping in these spaces. A suitable example confirms our results. We also present an application for a generalized class of nonlinear integral equations. Our results generalize and extend the results of Karapınar et al. (IEEE Access 7:89026–89032, 2019). [ABSTRACT FROM AUTHOR]

Let f1 and f2 be two orientation-preserving circle homeomorphisms with the same irrational rotation number ρ and each with a single break point b1 and b2, respectively. Suppose that the derivatives Df1 and Df2 satisfy a certain Zygmund condition except break points and the jumps σ(b1)=Df1(b1-0)Df1(b1+0), σ(b2)=Df2(b2-0)Df1(b2+0) do not coincide. Then the map ψ conjugating f1 and f2 is singular. [ABSTRACT FROM AUTHOR]

The aim of this paper is to extend some results of Feng and Liu , Klim and Wardowski , Ćirić and others from the context of metric spaces to cone metric spaces using the concept of sequentially lower semicontinuous. Examples are provided to illustrate the theory. [ABSTRACT FROM AUTHOR]

Let f be an orientation-preserving circle diffeomorphism with irrational rotation number and log f' be absolutely continuous, f"= f' ϵ Lp; p>1: By using the ratio distortion approach, we prove a sharper result for the key estimate of the Y. Katznelson and D. Ornstein theorem on the absolute continuity of the conjugacy for such circle diffeomorphism f . [ABSTRACT FROM AUTHOR]

We consider distribution results for closed orbits of the partially hyperbolic system: an ergodic toral automorphism à with respect to a (G, ρ)—extension A. In particular we obtain an analogue of the Chebotarev theorem in this situation which is an asymptotic formula for the number of closed orbits of the base transformation according to how they lift onto the extension space. To arrive at this result we introduce a cyclic extension  of A and deduce that  and A is essentially a group extension and homogeneous extension of à respectively. This observation of a group extension is similar to the setting previously studied by Parry & Pollicott and using the prime orbit theorem of Waddington we then derive at an auxiliary result for the group extension analogoues to Parry & Pollicott. Finally we relate this auxiliary result to the homogeneous extension by resorting to the work of Noorani & Parry. [ABSTRACT FROM AUTHOR]

The goal of this study is to offer an exclusive functional conversion to produce (n+1)-dimensional dual-mode nonlinear equations. This transformation has been implemented and new (3+1)-dimensional dual-mode Gradner-type and KdV-type have been established. Finally, the simplified bilinear method is used to tell the necessary conditions on these new models to have multiple singular-solitons. [ABSTRACT FROM AUTHOR]