Roque Sol, Marco A. (2006-08). A study of the sensitivity of topological dynamical systems and the Fourier spectrum of chaotic interval maps. Doctoral Dissertation.
Thesis
We study some topological properties of dynamical systems. In particular the rela- tionship between spatio-temporal chaotic and Li-Yorke sensitive dynamical systems establishing that for minimal dynamical systems those properties are equivalent. In the same direction we show that being a Li-Yorke sensitive dynamical system implies that the system is also Li-Yorke chaotic. On the other hand we survey the possibility of lifting some topological properties from a given dynamical system (Y, S) to an- other (X, T). After studying some basic facts about topological dynamical systems, we move to the particular case of interval maps. We know that through the knowl- edge of interval maps, f : I -> I, precious information about the chaotic behavior of general nonlinear dynamical systems can be obtained. It is also well known that the analysis of the spectrum of time series encloses important material related to the signal itself. In this work we look for possible connections between chaotic dynamical systems and the behavior of its Fourier coefficients. We have found that a natural bridge between these two concepts is given by the total variation of a function and its connection with the topological entropy associated to the n-th iteration, fn(x), of the map. Working in a natural way using the Sobolev spaces Wp,q(I) we show how the Fourier coefficients are related to the chaoticity of interval maps.
We study some topological properties of dynamical systems. In particular the rela- tionship between spatio-temporal chaotic and Li-Yorke sensitive dynamical systems establishing that for minimal dynamical systems those properties are equivalent. In the same direction we show that being a Li-Yorke sensitive dynamical system implies that the system is also Li-Yorke chaotic. On the other hand we survey the possibility of lifting some topological properties from a given dynamical system (Y, S) to an- other (X, T). After studying some basic facts about topological dynamical systems, we move to the particular case of interval maps. We know that through the knowl- edge of interval maps, f : I -> I, precious information about the chaotic behavior of general nonlinear dynamical systems can be obtained. It is also well known that the analysis of the spectrum of time series encloses important material related to the signal itself. In this work we look for possible connections between chaotic dynamical systems and the behavior of its Fourier coefficients. We have found that a natural bridge between these two concepts is given by the total variation of a function and its connection with the topological entropy associated to the n-th iteration, fn(x), of the map. Working in a natural way using the Sobolev spaces Wp,q(I) we show how the Fourier coefficients are related to the chaoticity of interval maps.