/ 93 03 01 8 v 1 3 1 M ar 1 99 3 NON - ERGODICITY FOR C 1 EXPANDING MAPS
نویسنده
چکیده
In this paper, we use a remarkable recent result of Bramson and Kalikow, [BK], to produce an example of a C expanding map of the circle which preserves more than one absolutely continuous probability measure. We first introduce some abbreviations and definitions. An expanding map of the circle is a differentiable map f from the circle to itself such that |Dxf | ≥ C for some fixed constants C > 1, n ∈ N, for all x ∈ S. Such a map must be an r-fold cover of the circle for some r ≥ 2. We will denote by E, the collection of C expanding maps of the circle, where k ≥ 1. Note that if k is non-integral, then we mean that the ⌊k⌋th derivative of the map is Hölder with exponent k − ⌊k⌋. An absolutely continuous invariant probability measure (abbreviated to ACIM) for a map T : S → S is a Borel probability measure μ such that μ(TB) = μ(B) for all Borel sets B and such that μ is absolutely continuous with respect to Lebesgue measure λ on S (that is λ(A) = 0 ⇒ μ(A) = 0). The example which we describe will make use of symbolic dynamics. The central object of the investigation will be the set Σs = {0, 1, . . . s − 1} Z + . This is endowed with the product topology by taking the metric
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تاریخ انتشار 1993