Simple permutations with order 4n+24n + 2. Part I

The problem of genealogy of permutations has been solved partially by Stefan (odd order) and Acosta-Hum\'anez & Bernhardt (power of two). It is well known that Sharkovskii's theorem shows the relationship between the cardinal of the set of periodic points of a continuous map, but simple permutations will show the behavior of those periodic points. This paper studies the structure of permutations of mixed order $4n+2$, its properties and a way to describe its genealogy by using Pasting and Reversing.Comment: 17 page

Weak index pairs and the Conley index for discrete multivalued dynamical systems. Part II: properties of the index

Motivation to revisit the Conley index theory for discrete multivalued dynamical systems stems from the needs of broader real applications, in particular in sampled dynamics or in combinatorial dynamics. The new construction of the index in [B. Batko and M. Mrozek, {\em SIAM J. Applied Dynamical Systems}, 15(2016), pp. 1143-1162] based on weak index pairs, under the circumstances of the absence of index pairs caused by relaxing the isolation property, seems to be a promising step towards this direction. The present paper is a direct continuation of [B. Batko and M. Mrozek, {\em SIAM J. Applied Dynamical Systems}, 15(2016), pp. 1143-1162] and concerns properties of the index defined therin, namely Wa\.zewski property, the additivity property, the homotopy (continuation) property and the commutativity property. We also present the construction of weak index pairs in an isolating block

Combinatorial dynamics

In Chapter 1, we consider many topics that are both combinatorial and dynamical in nature. In particular, we study substitution maps, subword complexity, symbolic dynamics and interval-exchange maps. After describing the basic concepts and notation, we study the subword complexity functions that arise from substitutions connected with /3-transformations. We then make some general observations regarding subword complexity functions associated with substitutions, before going on to study some specific examples with quadratic growth in section 1.4. In section 1.5, we study the symbolic dynamics associated with these types of substitutions, generalising the notions of recurrence, minimality etc. In section 1.6, we briefly describe and compute an invariant measure for the substitutions considered in section 1.4. We then prove a result that describes a connection between the symbolic dynamics and interval-exchange maps, and apply it to these substitution maps. In Chapter 2, we study a dynamical skew-product and some of the combinatorial questions that it raises. In sections 2.1 and 2.2 we describe the skew-product, and explain the connection between it and some one-player games. We then describe and analyse a code-word problem, and explain how we can generalise our results. In sections 2.6 and 2.7, we study a continuous version of the problem and prove a result that might shed some light on the original skew-product. At the end of both chapters, we present some problems which we believe to be still open, and suggest ideas for further research into the topics presented in this thesis

Period sets of linear toral endomorphisms on T2T^2

The period set of a dynamical system is defined as the subset of all integers $n$ such that the system has a periodic orbit of length $n$. Based on known results on the intersection of period sets of torus maps within a homotopy class, we give a complete classification of the period sets of (not necessarily invertible) toral endomorphisms on the $2$--dimensional torus $\mathbb{T}^2$.Comment: 10 page