Itineraries of rigid rotations and diffeomorphisms of the circle

We examine the itinerary of $0\in S^{1}=\R/\Z$ under the rotation by \alpha\in\R\bs\Q. The motivating question is: if we are given only the itinerary of 0 relative to $I\subset S^{1}$, a finite union of closed intervals, can we recover $\alpha$ and $I$? We prove that the itineraries do determine $\alpha$ and $I$ up to certain equivalences. Then we present elementary methods for finding $\alpha$ and $I$. Moreover, if $g:S^{1}\to S^{1}$ is a $C^{2}$, orientation preserving diffeomorphism with an irrational rotation number, then we can use the orbit itinerary to recover the rotation number up to certain equivalences.Comment: Added error estimates in response to referees' comment

The Period adding and incrementing bifurcations: from rotation theory to applications

International audienceThis survey article is concerned with the study of bifurcations of discontinuous piecewise-smooth maps, with a special focus on the one-dimensional case. We review the literature on circle maps and quasi-contractions and provide paths through this literature to prove sufficient conditions for the occurrence of two types of bifurcation scenarios involving rich dynamics. The first scenario consists of the appearance of periodic orbits whose symbolic sequences and “rotation” numbers follow a Farey tree structure; the periods of the periodic orbits are given by consecutive addition. This is called the period adding bifurcation, and the proof of its existence relies on results for maps on the circle. In the second scenario, symbolic sequences are obtained by consecutive attachment of a given symbolic block, and the periods of periodic orbits are incremented by a constant term. This is called the period incrementing bifurcation, and its proof relies on results for maps on the interval. We also discuss the expanding cases, as some of the partial results found in the literature also hold when these maps lose contractiveness. The higher-dimensional case is also discussed by means of quasi-contractions. We provide applied examples in control theory, power electronics, and neuroscience, where these results can be used to obtain precise descriptions of their dynamics

Dynamics on the boundary of Fatou components

Treballs finals del Màster en Matemàtica Avançada, Facultat de matemàtiques, Universitat de Barcelona, Any: 2021, Director: Núria Fagella Rabionet[en] The aim of this project is to compile the known results about the dynamics on the boundary of invariant simply-connected Fatou components, as well as the questions which are still open concerning the topic. We focus on ergodicity and recurrence. One of the main tools to deal with this kind of questions is to study the boundary behaviour of the associate inner functions. Therefore, the project is divided in two parts. Firstly, ergodicity and recurrence are studied for inner functions. Secondly, these results are applied to study the dynamics on the boundary of invariant simply-connected Fatou components. Moreover, we study the concrete example $f(z)=z+e^{-z}$, which presents infinitely many invariant doubly-parabolic Baker domains $U_{k}$. Making use of the associate inner function, which can be computed explicitly, we give a complete characterization of the periodic points in $\partial U_{k}$ and prove the existence of uncountably many curves of non-accessible escaping points