Notes on zygmund functions

In this paper we study a class of continuous functions satisfying a certain Zyg-mund condition dependent on a parameter γ > 0. It shown that the modulus of continuity of such functions is O(δ(log 1/δ)1-γ) if ∈ (0, 1) and O(δ(log log 1/δ )) if γ = 1. Moreover, these functions are differentiable if γ > 1. These results extend the results in literatures [4], [5]

Circle homeomorphisms with two break points.

Abstract Let f i ∈ C 2+α (S 1 \{a i , b i }), α > 0, i = 1, 2, be circle homeomorphisms with two break points a i , b i i.e. discontinuities in the derivative Df i , with identical irrational rotation number ρ and , where µ i are the invariant measures of f i , i = 1, 2. Suppose, the products of the jump ratios of Df 1 and Df 2 do not coincide, i.e. Df2(b2+0) . Then the map ψ conjugating f 1 and f 2 is a singular function, i.e. it is continuous on S 1 , but Dψ(x) = 0 a.e. with respect to Lebesgue measure

The existence of ϒ-fixed point for the multidimensional nonlinear mappings satisfying (ψ, θ, ϕ)-weak contractive conditions

In this paper we prove the existence of ϒ-fixed point for a multidimensional nonlinear mappings F : Xk → X defined on the partially ordered metric spaces and satisfying (ψ, θ, ϕ)-weak contractive conditions. Moreover, we prove the uniqueness of that fixed point under extra conditions to (ψ, θ, ϕ)-weak contractive conditions

Estimates on the number of orbits of the Dyck shift

In this paper, we get crucial estimates of fundamental sums that involve the number of closed orbits of the Dyck shift.These estimates are given as the prime orbit theorem, Mertens’ orbit theorem, Meissel’s orbit theorem and Dirichlet series. Different and more direct methods are used in the proofs without any complicated theoretical discussions

Counting Closed Orbits for the Dyck Shift

The prime orbit theorem and Mertens’ theorem are proved for a shift dynamical system of infinite type called the Dyck shift. Different and more direct methods are used in the proof without any complicated theoretical discussion

On conjugations of circle homeomorphisms with two break points

Let $f_i\in C^{2+\alpha}(S^1\setminus \{a_i,b_i\}), \alpha >0, i=1,2$ be circle homeomorphisms with two break points $a_i,b_i$, i.e. discontinuities in the derivative $f_i$, with identical irrational rotation number $rho$ and $\mu_1([a_1,b_1])= \mu_2([a_2,b_2])$, where $\mu_i$ are invariant measures of $f_i$. Suppose the products of the jump ratios of $Df_1$ and $Df_2$ do not coincide, i.e. $\frac{Df_1(a_1-0)}{Df_1(a_1+0)}\times \frac{Df_1(b_1-0)}{Df_1(b_1+0)}\neq \frac{Df_2(a_2-0)}{Df_2(a_2+0)}\times \frac{Df_2(b_2-0)}{Df_2(b_2+0)}$. Then the map $\psi$ conjugating $f_1$ and $f_2$ is a singular function, i.e. it is continuous on $S^1$, but $D\psi = 0$ a.e. with respect to Lebesgue measureComment: 16 pages, 2 figures, to appear in Ergodic Theory and Dynamical System