The finitary isomorphism theorem, due to Keane and Smorodinsky, raised the natural question of how "finite" the isomorphism can be, in terms of moments of the coding radius. More precisely, for which values does there exist an isomorphism between any two i.i.d. processes of equal entropy, with coding radii exhibiting finite t-moments? [3, 4]. Parry [13] and Krieger [10] showed that those finite moments must be lesser than 1 in general, and Harvey and Peres [5] showed that they must be lesser than 1/2 in general. However, the question for the range between 0 and 1/2 remained open, and in fact no general construction of an isomorphism was shown to exhibit any non trivial finite moments. In the present work we settle this problem, showing that between any two aperiodic Markov processes (and i.i.d. processes in particular) of the same entropy, there exists an isomorphism f with coding radii exhibiting finite t-moments for all t in (0,1/2). The isomorphism is constructed explicitly, and the tails of the radii are shown to be optimal up to a poly-logarithmic factor.

翻译：Keane与Smorodinsky提出的有限同构定理自然引出一个问题：同构的"有限性"程度如何，即编码半径的矩表现如何？更精确地说，对于哪些t值，任意两个等熵的独立同分布过程之间存在编码半径具有有限t阶矩的同构？[3, 4] Parry [13]与Krieger [10]证明这些有限矩通常必须小于1，而Harvey与Peres [5]证明其通常必须小于1/2。然而，对于0到1/2区间的问题仍然悬而未决，实际上尚未有构造出的同构被证明具有任何非平凡的有限矩。本研究解决了该问题，证明在任意两个具有相同熵的非周期马尔可夫过程（特别是独立同分布过程）之间，存在一个同构f，其编码半径对所有t∈(0,1/2)均具有有限t阶矩。该同构被显式构造，且其半径尾部分布被证明在多项式对数因子内达到最优。