In one complex variable dynamics, Sullivan’s theorem ([6]) gives a complete classification of the Fatou components that can appear for a rational map f . Consequently we can have only periodic components U and U can be one of the following: 1. U attracting basin; 2. U parabolic domain; 3. U Siegel disk; 4. U Herman ring. Cases 3. and 4. are called rotation domains. In these cases, the rational ...

We establish an approximation of the activity current Tc in the parameter space of a holomorphic family f of rational functions having a marked critical point c by parameters for which c is periodic under f , i.e., is a superattracting periodic point. This partly generalizes a Dujardin–Favre theorem for rational functions having preperiodic points, and refines a Bassanelli–Berteloot theorem on ...

Counting the number of distinct factors in the words of a language gives a measure of complexity for that language similar to the factor-complexity of infinite words. Similarly as for infinite words, we prove that this complexity functions f(n) is either bounded or f(n) ≥ n+1. We call languages with bounded complexity periodic and languages with complexity f(n) = n + 1 Sturmian. We describe the...

where φp(x) = |x|p–x for x =  and p > ; σ and c are given constants with |c| = ; φp() = , f () = . The conjugate exponent of p is denoted by q, i.e.  p +  q = . f , g , β , e, and τ are real continuous functions on R; τ , β , and e are periodic with periodic T , T >  is a constant; ∫ T  e(t)dt = , ∫ T  β(t) = . As we know, the p-Laplace Rayleigh equation with a deviating argumen...

We investigate the iterative behaviour of continuous order preserving subhomogeneous maps f :K→K, where K is a polyhedral cone in a finite dimensional vector space. We show that each bounded orbit of f converges to a periodic orbit and, moreover, the period of each periodic point of f is bounded by βN = max q+r+s=N N ! q!r!s! = N ! ⌊ N 3 ⌋ ! ⌊ N + 1 3 ⌋ ! ⌊ N + 2 3 ⌋ ! ∼ 3 N +1 √ 3 2πN , where ...

We obtain sufficient conditions for the existence of three T -periodic solutions of the first order functional differential equation u(t) = a(t)g(u(t))u(t) − b(t)f (u(t − τ(t))), where a, b, τ ∈ C(R, R) are T -periodic functions, f , g ∈ C(R, R), and g is not necessarily bounded. As an application of our theorem, we also derived criteria for the existence of three T -periodic solutions of the e...

It is known that two commuting continuous functions on an interval need not have a common fixed point. However, it is not known if such two functions have a common periodic point. We had conjectured that two commuting continuous functions on an interval will typically have disjoint sets of periodic points. In this paper, we first prove that S is a nowhere dense subset of [0,1] if and only if {f...

Let T be a tree with n vertices. Let f : T → T be continuous and suppose that the n vertices form a periodic orbit under f . The combinatorial information that comes from possible permutations of the vertices gives rise to an irreducible representation of Sn. Using the algebraic information it is shown that f must have periodic orbits of certain periods. Finally, a family of maps is defined whi...

We study strong solutions u : R → X, a Banach space X, of the nth-order evolution equation u(n) −Au(n−1) = f , an infinitesimal generator of a strongly continuous group A : D(A) ⊆ X → X, and a given forcing term f : R → X. It is shown that if X is reflexive, u and u(n−1) are Stepanov-bounded, and f is Stepanov almost periodic, then u and all derivatives u′, . . . ,u(n−1) are strongly almost per...

We study the map 9: C!C defined by 9(w, z)=(z, z+w) and the associated collection of sequences [zj] satisfying the recurrence zj+2=zj+1+zj . Iteration of 9 is equivalent to the study of such sequences. We analyze growth rates of the sequences, positive sequences, periodic and asymptotically periodic sequences and establish the existence of doubly infinite homoclinic sequences, non-zero sequence...