ABSTRACT We study a skew product IFS on the cylinder defined by Baker-like maps associated to a finite family of potential functions and the doubling map. We show that there exist a compact invariant set with attractive behaviour and a random SRB measure whose support is in that set. We also study the IFS ergodic optimization problem for that finite family of potential functions and characterize the maximizing measures and the critical value through a discounting limit. This shows the connection between this maximization problem and the superior boundary of the compact invariant set, which is described as a graph of the solution of the Bellman equation.

中文翻译：

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关于斜积 IFS 与有限势族遍历优化之间的联系

摘要 我们研究了圆柱体上的偏斜积 IFS，该圆柱体由与有限势函数族和倍增映射相关的 Baker 类映射定义。我们表明存在一个具有吸引力行为的紧凑不变集和一个支持在该集合中的随机 SRB 度量。我们还研究了该有限潜在函数族的 IFS 遍历优化问题，并通过贴现限制来表征最大化度量和临界值。这显示了这个最大化问题和紧致不变集的上边界之间的联系，它被描述为贝尔曼方程的解图。