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In this paper, we propose a numerical scheme of the predictor-corrector type for solving nonlinear fractional initial value problems, the chosen fractional derivative is called the Atangana-Baleanu derivative defined in Caputo sense (ABC). This proposed method is based on Lagrangian quadratic polynomials to approximate the nonlinearity implied in the Volterra integral which is obtained by reducing the given fractional differential equation via the properties of the ABC-fractional derivative. Through this technique, we get corrector formula with high accuracy which is implicit as well as predictor formula which is explicit and has the same precision order as the corrective formula. On the other hand, the so-called memory term is computed only once for both prediction and correction phases, which indicates the low cost of the proposed method. Also, the error bound of the proposed numerical scheme is offered. Furthermore, numerical experiments are presented in order to assess the accuracy of the new method on two differential equations. Moreover, a case study is considered where the proposed predictor-corrector scheme is used to obtained approximate solutions of ABC-fractional generalized SI (susceptible-infectious) epidemic model for the purpose of analyzing dynamics of the suggested system as well as demonstrating the effectiveness of the new method to solve systems dealing with real-world problems.

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 space. 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 analiticity of the Artin-Mazur zeta function of the space. Despite having a closed-form expression, the zeta function is expressed implicitly in terms of several signed subset matrices, and this makes the study on its analyticity to be seemingly difficult. In this paper, we will prove the asymptotic behaviours of the counting functions for a sofic shift via its zeta function. This involves investigating the properties of the said matrices. Suprisingly, the proof is rather short and only uses well-known facts about a sofic shift, especially on its minimal right-resolving presentation., Comment: 11 pages

Let $f_{i},$ $i=1,2$ be piecewise-smooth $C^{1}$ circle homeomorphisms with two break points, $\log Df_{i},$ $i=1,2$ are absolutely continuous on each continuity intervals of $Df_{i}$ and $D\log Df_{i}\in L^{p}$ for some $p>1.$ Suppose, the jump ratios of $f_{1} $ and $f_{2} $ at their break points do not coincide but have the same total jumps (i.e. the product of jump ratios) and identical irrational rotation number of bounded type. Then the conjugation $h$ between $f_{1} $ and $f_{2} $ is a singular function, i.e. it is continuous on $S^1,$ but $Dh(x)=0$ a.e. with respect to Lebesgue measure., 21 pages 2 figures

This thesis consists of four chapters. Chapters 1 and 2 are somewhat related in the sense that they deal with similar dynamical systems. Each chapter comes complete with its own references and notations.\ud For the convenience of the reader, we provide an introduction and indeed an elongated summary to the whole thesis in Chapter 0.\ud \ud In Chapter 1, we study how closed orbits of a subshift of finite type hits to a finite homogeneous extension. In particular, we obtain an asymptotic formula for the number of closed orbits according to how they lift to the extension space. We apply our findings to the case of finite extensions and also to automorphism extensions of hyperbolic toral automorphisms.\ud \ud Chapter 2 deals with lifting ergodic properties of an arbitrary measure preserving transformation T to homogeneous extensions of T. Our results extends well known theorems already obtained for the case of compact group extensions of measure-preserving transformations. We also give simplified results to the special case when the base transformation is a Markov shift and the skewing-function depends on a finite number of coordinates.\ud \ud In Chapter 3, we look at the rate of mixing of rectangle sets of two dimensional Markov shifts with respect to the natural shift actions. We show that if one of the matrix defining the Markov measure is aperiodic then this rate is exponentially fast. We provide an example to illustrate what could happen in general.