Random matrix products of 2 x 2 matrices may be thought of in a dynamical system setting as iterations functions of mobius maps, with matrix multiplication replacing composition of functions. Sometimes branches of mobius maps may be restricted to form classical dynamical systems. One such family is the tent family about which much is known. Similar properties are investigated for a two parameter family of symmetric piecewise mobius snaps (which include the tent family). Using kneading theory, symbolic dynamics, and other techniques, A parameter space is found which foliates into curves of constant kneading sequence on which maps may be pairwise conjugate depending on if the maps restricted to a (forward invariant) core interval ire transitive. Investigations into iterated function systems given by the inverse branches of the symmetric family are made by defining a shift on two symbols (depending upon some interval of definition) that models the iterated function system. Continuous deformations of the interval are made and properties of entropy are found. In some cases entropy of the shift space is continuous as the interval is deformed, while in other cases there are discontinuities.
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机译：在动力学系统设置中，可以将2 x 2矩阵的随机矩阵乘积视为mobius映射的迭代函数，用矩阵乘法代替函数的组合。有时，莫比乌斯图的分支可能会受到限制以形成经典的动力学系统。帐篷家庭就是其中之一，众所周知。对于对称分段莫比乌斯按扣的两个参数族（包括帐篷族），研究了相似的特性。使用揉合理论，符号动力学和其他技术，发现了一个参数空间，该参数空间形成了恒定的揉合序列曲线，在该曲线上，映射可能成对共轭，取决于映射是否局限于（向前不变的）核心区间，这些映射是可传递的。由对称族的反向分支给出的迭代函数系统的研究是通过对两个对迭代函数系统建模的符号的移位（取决于定义的某个间隔）进行定义的。进行区间的连续变形并发现熵的性质。在某些情况下，随着间隔的变形，移位空间的熵是连续的，而在其他情况下，则存在不连续性。
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