凸单调齐次映射的谱定理与遍历控制
Spectral Theorem for Convex Monotone Homogeneous Maps, and Ergodic Control
摘要：
我们考虑单调的凸映射f：R^n -> R^n（即，保持R^n）的乘积序，并且对于超范数是非扩张的。这包括加性齐次的凸单调映射（即，这与常数的加法交换）。证明了当f非空时，f的不动点集同构于R^n的凸下半格，其维数至多等于由f的切仿射映射定义的临界图的强连通分支数.这产生特别是一个独特的结果，遍历控制问题的偏置向量。这推广了Lanery，Romanovsky，Schweitzer和Federgruen以前得到的结果，遍历控制问题的有限状态和动作空间，对应于特殊情况下的分段仿射映射f。我们还证明了f的周期轨道的长度受其临界图的圈性的限制，这意味着f的可能轨道长度恰好是n个字母上的对称群的元素的阶.
英文摘要：
We consider convex maps f:R^n -> R^n that are monotone (i.e., that preserve the product ordering of R^n), and nonexpansive for the sup-norm. This includes convex monotone maps that are additively homogeneous (i.e., that commute with the addition of constants). We show that the fixed point set of f, when it is non-empty, is isomorphic to a convex inf-subsemilattice of R^n, whose dimension is at most equal to the number of strongly connected components of a critical graph defined from the tangent affine maps of f. This yields in particular an uniqueness result for the bias vector of ergodic control problems. This generalizes results obtained previously by Lanery, Romanovsky, and Schweitzer and Federgruen, for ergodic control problems with finite state and action spaces, which correspond to the special case of piecewise affine maps f. We also show that the length of periodic orbits of f is bounded by the cyclicity of its critical graph, which implies that the possible orbit lengths of f are exactly the orders of elements of the symmetric group on n letters.
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