Research On Non-renormalizable Bimodal Maps

Polynomial dynamical system is one of the main topics of one-dimensional dynamical system.We can study polynomials as interval maps,as well as complex maps,and we consider the properties of Julia sets of them.From the perspective of topology,KozlovskiShen-van strien proved the real Fatou conjecture,they showed the hyperbolic system in real polynomials is dense.However,there are few results on how to understand polynomial dynamics from the perspective of probability,which is also the frontier of onedimensional dynamic system.For the family of real analytic unimodal maps,Avila,Lyubich,de Melo and Shen proved that almost all mappings are either hyperbolic or have absolutely continuous invariant measures.For the family of real polynomials,the key to this problem lies in the study of the existence of wild attractors and invariant measures for non-renormalizable multimodal maps.For cubic polynomials,Swiatiek-Vargas reveal that their geometric properties are different from quadratic polynomials,which indicates that the measure properties of cubic polynomials will be more complex.At present,most researches on cubic polynomials require that the two critical points are relatively independent.There are few related studies on the non-renormalizable cubic polynomials with two recurrent critical points which are contained in the same minimal Cantor set.The difficulties of this kind of research is that we can not ignore the interaction of critical points when dealing with multimodal maps using the ’principal nest’,so we face technical challenges in distortion estimation.Recently,Vargas characterized the Fibonacci combinatorics in bimodal maps by considering the first return maps in a more symmetrical way.We call the nice return domains as twin principal nest.Compared with the principal nest of multimodal maps,the first return maps on twin principal nest are closer to the polynomial-like maps in the sense of Douady-Hubbard.For bimodal maps with recurrent critical points,we can define their twin principal nest.From the viewpoint of generalized renormalization,we characterize the combinatorial properties of bimodal maps from the return times on the return domains intersecting the critical points orbits,and the basic types of the first return maps.Firstly,we prove that there is a bimodal map with such a combinatorial sequence if {(θn,rn,tn)｝n≥1 satisfies the admissibility condition.Using the rigidity theorem,we can find such bimodal map in the family of real cubic polynomials.We prove that the scaling factor of the bimodal map in this family decreases at least exponentially.These maps have no Cantor attractor,but they admit acip.Our strategy is to expand the interval map into a generalized box map,and then study the asymptotic properties of the moduli of its separating annuli.We will use the Yoccoz puzzle to get complex a priori bounds since the real bound theorem does not hold under our setting.Note that the construction of the Yoccoz puzzle depends on B ottcher theorem,this is the reason why our results are about cubic polynomials.In the unimodal case,the decay of the scaling factor of principal nest is closely related to the measure property of the map.We hope to continue this research in the bimodal case.Using kneading map,we construct a Markov map and show that it is semi-conjugate with the map induced by Fibonacci bimodal map.We express the convergence rate of the scaling factor in a geometric manner,and construct a single parameter piecewise linear model fλ,whose critical index I can be specifically expressed as 3+2log(1-λ)/logλ.Moreover,we discuss the phase transition of measure properties when the parameter changes:Ifλ∈ G(1/2,1).that is,l>5,then fλ has a wild attractor;If A∈((?),1/2)，i.e.4<l<5,then fλ has no wild attractor,but has an infinite σ-finite acim;If A∈(0,(?)),that is,3<l<4,then fλ has an acip.This thesis is organized as follows:In Chapter 1,we firstly review the origin of one-dimensional dynamical systems,then introduce the research background related to this thesis,and state the main results of this thesis.In Chapter 2,we introduce some basic concepts and known results about interval mapping,ergodic theory and complex dynamical systems.In Chapter 3,use the viewpoint of generalized renormalization,we characterize the combinatorial properties of bimodal maps from the return times on the return domains intersecting the critical points orbits,and the basic types of the first return maps.Then,by studying the asymptotic properties of the moduli of its separating annuli,we obtain the ’decay of geometric’ phenomenon of a class of real cubic polynomials,and prove that this kind of cubic polynomials have no Cantor attractor but admit acip by using random walk.In Chapter 4,we construct a Markov map using kneading map and show that it is semi-conjugate with the map induced by Fibonacci bimodal map.We express the convergence rate of the scaling factor in a geometric manner,and construct a single parameter piecewise linear model fλ.Moreover,we discuss the phase transition of measure properties when the parameter changes.