COMBINATORIAL DYNAMICS OF STRIP PATTERNS OF QUASIPERIODIC SKEW PRODUCTS IN THE CYLINDER

Abstract

The thesis consists of two parts. In the first we aim at extending the results and techniques from Fabbri et al. 2005 to study the Combinatorial Dynamics, the > and the topological Entropy of certain quasiperiodically forced skew-product on the cylinder. This theory gives a structured demonstration from the Sharkovski Theorem as a corollary, proved initially in Fabbri et al., 2005. About entropy defines the notion of horseshoe in this context and shwow, as in the interval case, if one of these functions has a s-horseshoe then its topological entropy is greater than or equal to log s. It follows lower entropy based on periodic orbits periods. This represents an similar extension to the results a l'interval in this context._x000D_ _x000D_ In the above context arises naturally the following question: Sharkovsky theorem holds restricted curves instead of bands general? The aim of the second part of the report is to answer this question negatively by a contraexample: It constructs a function that has two curves as periodic orbit of period 2 (which are, in fact, the upper and lower circles cylinder) with no invariant curve (only has an invariant pseudo-curve). In particular, this shows that there are quasiperiodically forced skew-product on the cylinder without invariant curves. This is the first analytical result of this kind appearing in the literature despite the existence of previous numerical evidence in this regard._x000D_ _x000D_ The results are only the first stage in understanding analytic/topological dynamics of these applications, which work via a future job.

Date of Award	26 May 2016
Original language	Undefined/Unknown
Supervisor	Lluis Alseda Soler (Director) & Francesc Mañosas Capellades (Director)