Roots of continuous piecewise monotone maps of an interval.
We show that the Sharkovskiĭ ordering of periods of a continuous real function is also valid for every function with connected graph. In particular, it is valid for every DB₁ function and therefore for every derivative. As a tool we apply an Itinerary Lemma for functions with connected graph.
We study 1) the slopes of central branches of iterates of S-unimodal maps, comparing them to the derivatives on the critical trajectory, 2) the hyperbolic structure of Collet-Eckmann maps estimating the exponents, and under a summability condition 3) the images of the density one under the iterates of the Perron-Frobenius operator, 4) the density of the absolutely continuous invariant measure.
We show that the theorem proved in [8] generalises the previous results concerning orientation-preserving iterative roots of homeomorphisms of the circle with a rational rotation number (see [2], [6], [10] and [7]).
Let ϕ be an arbitrary bijection of . We prove that if the two-place function is subadditive in then must be a convex homeomorphism of . This is a partial converse of Mulholland’s inequality. Some new properties of subadditive bijections of are also given. We apply the above results to obtain several converses of Minkowski’s inequality.
We consider the dynamical system (𝒜, Tf), where 𝒜 is a class of differential real functions defined on some interval and Tf : 𝒜 → 𝒜 is an operator Tfφ := fοφ, where f is a differentiable m-modal map. If we consider functions in 𝒜 whose critical values are periodic points for f then, we show how to define and characterize a substitution system associated with (𝒜, Tf). For these substitution systems, we compute the growth rate of the...
We consider the functional equation where is a given increasing homeomorphism of an open interval and is an unknown continuous function. In a series of papers by P. Kahlig and J. Smítal it was proved that the range of any non-constant solution is an interval whose end-points are fixed under and which contains in its interior no fixed point except for . They also provide a characterization of the class of monotone solutions and prove a necessary and sufficient condition for any solution...
We consider the functional equation where is a given homeomorphism of an open interval and is an unknown continuous function. A characterization of the class of continuous solutions is given in a series of papers by Kahlig and Smítal 1998–2002, and in a recent paper by Reich et al. 2004, in the case when is increasing. In the present paper we solve the converse problem, for which continuous maps , where is an interval, there is an increasing homeomorphism of such that . We...
We show that the Covering Principle known for continuous maps of the real line also holds for functions whose graph is a connected subset of the plane. As an application we find an example of an approximately continuous (hence Darboux Baire 1) function f: [0,1] → [0,1] such that any closed subset of [0,1] can be translated so as to become an ω-limit set of f. This solves a problem posed by Bruckner, Ceder and Pearson [Real Anal. Exchange 15 (1989/90)].