Combinatorial dynamics on the interval and a generalization of Sharkovskiĭ's theorem

We study the discrete one dimensional dynamical systems given by continuous functions mapping a closed real interval into itself and the law of coexistence of periodic orbits for such systems. In chapter 1 we study invariant measures for a continuous function which maps a real interval into itself. We show that the ratio of the measures of the two subintervals into which it is divided by a fixed point is constrained by the set of periods of periodic points. As a consequence of this we get new information about the law of coexistence of periodic orbits. In chapter 2 we study the law of coexistence of different types of periodic orbits more closely. Based on the idea from chapter 1 we introduce the term eccentricity of a periodic orbit and study the coexistence law between periodic orbits with different eccentricities. We also characterize those periodic orbits with a given eccentricity that are simplest from the point of view of the coexistence law. We obtain a generalization of Sharkovskii's Theorem where the notion of period of periodic orbit is replaced by the notion of eccentricity of periodic orbit. Chapter 2 is independent of chapter 1 but uses ideas that originated in the work covered by chapter 1

X-minimal patterns and a generalization of Sharkovskiĭ's theorem

We study the law of coexistence of different types of cycles for a continuous map of the interval. For this we introduce the notion of eccentricity of a pattern and characterize those patterns with a given eccentricity that are simplest from the point of view of the forcing relation. We call these patterns X-minimal. We obtain a generalization of Sharkovskiĭ's Theorem where the notion of period is replaced by the notion of eccentricity