Renormalizations of Unimodal and Bimodal Maps with Low Smoothness.

Abstract

Renormalization theory plays a key role in describing the dynamics of a given system at a small spatial scale by an induced dynamical system in the same class. The concept of renormalization arises in many forms though Mathematics and Physics. Period doubling renormalization operator was introduced by M. Feigenbaum and by P. Coullet and C. Tresser, to study asymptotic small scale geometry of the attractor of one dimensional systems which are at the transition from simple to chaotic dynamics. In this thesis, we discuss the renormalizations of unimodal maps and symmetric bimodal maps whose smoothness is C1+Lip, just below C2. In the rst section of the thesis, we explore the period tripling and period quintupling renormalizations below C2 class of unimodal maps. In the context of period tripling renormalization, there exists only one valid period tripling combinatorics. For a given proper tri-scaling data, we construct a nested sequence of a ne pieces whose end-points lie on the unimodal map and shrinking down to the critical point. Consequently, we prove the existence of a renormalization xed point, namely fs ; in the space of piece-wise a ne maps which are period tripling in nitely renormalizable, corresponding to a proper tri-scaling data s . Furthermore, we show that the renormalization xed point fs is extended to a C1+Lip unimodal map with a quadratic tip, by considering the period tripling combinatorics. Moreover, this leads to the fact that the period tripling renormalizations acting on the space of C1+Lip unimodal maps has unbounded topological entropy. Finally, by considering a small variation on the scaling data we show the existence of an another xed points of renormalization in the space of C1+Lip unimodal maps. This shows the contin- uum of xed points of period tripling renormalization operator. In the context of period quintupling renormalization, we have only three valid period quintupling combinatorics. For each combinatorics, there exists a piece-wise a ne xed point of renormalization operator, namely gs , corresponding to a proper quint-scaling data s : We show that the geometry of the invariant Cantor set of the map gs is more complex than the geometry of the invariant Cantor set of fs . Next, we describe the extension of gs to a C1+Lip space, and the topological entropy of period quintupling renormalization operator. The continuum's result also holds for this case. Consequently, we observe that the incre- ment of periodicity of renormalization increases the number of the possible combinatorics and each of them leads to such construction of renormalization xed point The second section of the thesis studies the renormalization of symmetric bimodal maps with low smoothness. The renormalization operator R is a pair of period tripling renormalization op- erators Rl and Rr which are de ned on piece-wise a ne period tripling in nitely renormalizable maps corresponding to a proper scaling data sl and sr, respectively. We prove the existence of the renormalization xed point in the space C1+Lip symmetric bimodal maps. Moreover, we show that the topological entropy of the renormalization operator de ned on the space of C1+Lip symmetric bimodal maps is in nite. Further, we prove the existence of a continuum of xed points of renormal- ization. The main result lead to the fact that the non-rigidity of the Cantor attractors of in nitely renormalizable symmetric bimodal maps, whose smoothness is below C2.