Parameter Choice for Families of Maps with Many Critical Points

We consider families of smooth one-dimensional maps with several critical points and outline the main ideas of the construction in the parameter space that allows to get infinite Markov Partitions and SRB measures for a positive set of parameter values. The construction is based on the properties of uniformly scaled Markov partitions from [6]. The same approach works for families of Henon-like maps. 1. FORMULATION OF THE THEOREM 1.1. There are several approaches to the construction of absolutely continuous invariant measures in families of non-hyperbolic maps depending on the parameter. The original method of [4] provides infinite Markov partitions for the related family of piecewise continuous expanding iterates Ft | ∆i = f ni t of the initial smooth map ft . The method of [1], [2], see also [10], [3] and subsequent papers, based on large deviations is non-Markov. It results in an a priori smaller set of parameter values satisfying Collet-Eckmann type conditions. The method of [13], [12] combines Markov property and Collet-Eckmann conditions. For further references see a recent survey [9] which in particular contains a detailed discussion of parameter exclusion. In any method the choice of parameter is an important part of the construction. Here we outline several features of a method based on the uniform scaling of Markov partitions, see [6]. There are similarities and differences between constructions in dimensions one and two. For example, distortion estimates in dimension two are more complicated, see [7], [11]. In the method based on uniformly scaled Markov partitions the problem of parameter choice for two-dimensional families is similar to the problem for one-dimensional families with several critical points . We mostly discuss that one-dimensional case and at the end outline specifics of the 2-dimensional construction. 1.2. After some preliminary construction, which includes transition to a first return map and taking several iterates of that map, see [5], [6], we get a family of one-dimensional C2 mappings Ft depending on the parameter t ∈ T0 = [t0,t1] with the following properties. For each t, Ft is piecewise continuous with a finite number of branches. The union of the domains of these branches is an interval I which can be considered independent of t , say I = [0,1] for all t. The branches of Ft are of three types. (1) There are m critical branches hl, l = 1, . . .m, whose domains are called central domains . Each central domain δl contains a single critical point Ol of Ft . Without loss of generality one can assume that Ol do not depend on t and so for l = 1, . . .m and for all t we have hlx(Ol) = 0 (2) Monotone expanding branches which we also call good branches (1) fi : ∆i → I 1