Research On Fibonacci-like Maps

In this thesis,we consider a class of unimodal maps with special combinatorics de-scribed by its principal nest.This class of maps is generated from the classic Fibonacci unimodal map.We aim to study the metric properties and universality law for this class.The dynamic property of Fibonacci interval maps has received a lot of interest.It has turned out that the geometric and statistical property of a Fibonacci unimodal map essentially depends on the critical order;when the critical order is less that 2+? for some ?>0 small,a Fibonacci map will have absolutely continuous invariant probability measure;when the critical order grows,the absolutely continuous invariant probability measure disappears,then the Fibonacci map will have a conservative absolutely contin-uous invariant ?-finite measure;if the critical order is sufficiently large,then the map will have a wild attractor,which implies that there is no absolutely continuous invariant probability measure.Let I0(?)I1(?)…(?)In(?)…be the principal nest of a unimodal map f.Consider the return domains and first return map gn to In.We only consider the return domains that intersect the orbit of the critical point.It is a fact that a unimodal map is of Fibonacci type if and only if:the return domains of In that intersect ?(c)consist of two components,one contains the critical point such that gn+1 restricted on it equals to gn2;gn+1 restricted on the other component equals to gn.We consider a class of unimodal maps that are generated from the classic Fibonacci map.For such a map,we will assume that the return domains of In that intersect ?(c)consist of two components.One of the components contains the critical point such that gn+1 restricted on it equals to gnpn;gn+1 restricted on the other component equals to gnqn.We will use the sequence of integer pairs?={(pn,qn)}n?1 to describe the combinatorics of f,called the combinatorial sequence of f.In our settings,the Fibonacci unimodal map has combinatorial sequence {(pn,qn)}n?1 satisfying p,? 2,qn?1.We first develop the Admissibility condition on ? which guarantees the existence of these maps and show that this condition is sufficient and necessary.Moreover,for maps with so-called 'bounded combinatorics',we prove that they possess bounded geometry when the critical order l is sufficiently large and hence admit no absolutely continuous invariant probability measure.For maps with a reluctantly recurrent critical point,we show that they display 'decay of geometry' and hence admit an absolutely continuous invariant probability measure.Although these maps are non-renormalizable,they can be renormalized in the sense of‘generalized renormalization'.By rescaling the first return map,we define the Fibonacci-like renormalization operator R.Instead of unimodal maps,we consider a new class of maps F:each f ? F is defined on the union of two disjoint open intervals I0,I1 has a unique critical point c ? I0 and maps each of its domain to a bigger interval.If there exists an integer k such that fl(c),…,f k(c)? I1 but f k+1(c),f k+2(c)?I0,then the first return map to I0,denoted by f1,still belongs to class F.Moreover,f1 equals to fk+1 on its central branch,and equals to f on its off-branch.Such a map is called Fibonacci-like renormalizable,k is called its renormalization period.Rescal-ing f1 to the original scale,we obtain the renormalization of f,denoted by Rf.We mainly consider infinitely Fibonacci-like renormalizable maps with bounded(all even or all odd)combinatorics and prove the universal limit for the renormalization sequence.Moreover,we establish the existence of a global attractor for the Fibonacci-like renor-malization operator with a 'horseshoe' structure.Finally,for infinitely Fibonacci-like renormalizable maps with stationary even combinatorics,we consider the fixed map f*under R.We show that the fixed map under the analytic version of Fibonacci-like renormalization operator is a hyperbolic fixed point in an appropriate Banach space with codimension one stable set.This thesis is organized as follows:In Chapter 1,we briefly recall the origin,development and main research contents in one-dimensional dynamical systems.Then we introduce the background and related research of our research.We also state the main results of this thesis.In Chapter 2,We provide background material,mathematical tools and known re-sults from one-dimensional dynamical systems,ergodic theory and complex dynamics.In Chapter 3,we study a class of non-renormalizable unimodal maps with spe-cial combinatorics described by its principal nest.We give the sufficient and necessary condition for the existence of this class of maps.Moreover,for maps with‘bounded combinatorics',we prove that they have no absolutely continuous invariant probability measure when the critical order l is sufficiently large;for maps with a reluctantly re-current critical point,we prove they have an absolutely continuous invariant probability measure.In Chapter 4,we define the Fibonacci-like renormalization operator R on a class of maps which are generated from the first return maps of Fibonacci-like maps.We mainly consider maps with the so-called‘bounded(even or odd)combinatorics'.We show that the orbit of each map from this class converges to a universal limit under iterates of the Fibonacci-like renormalization operator.We also construct the 'horseshoe' structure for this operator.In Chapter 5,we extend the Fibonacci-like renormalization operator R on a class of maps with stationary even combinatorics to a real analytic operator on an appropriate Banach space.We give a proof of the hyperbolicity of the corresponding fixed point under the Fibonacci-like renormalization operator using the local dynamics near the fixed point in this Banach space.