A Computer Verification for the Value of the Topological Entropy for Some Special Subshifts in the Lexicographical Scenario

Lorenz Attractor has been a source for many mathematical studies. Most of them deal with lower dimensional representations of its first return map. An one dimensional scenario can be modeled by the standard two parameter family of contracting Lorenz maps. The dynamics, in this case, can be modeled by a subshift in the Lexicographical model. These subshifts are the maximal invariant set for the shift map in some interval. For some of them the extremes, of the interval, are a minimal periodic sequence and a maximal periodic sequence which is an iteration of the lower extreme (by the shift map). For some of these subshifts the topological entropy is zero. In this case the dynamics (of the respective Lorenz map) is simple. Associated to any of these subshifts (let call it Λ) we consider an extension (let call it Γ) of it that contains Λ which also can be constructed by using an interval whose extremes can be defined by the extremes of Λ. For these extensions, we present here a computer verification of a result that compute its topological entropy. As a consequence, we can affirm: the longer the period of the periodic sequence is, then the lower complexity in the dynamics of the extension the associated map has. 2000 AMS Classification: Primary 37B10, 37B40, Secondary 37E05.