论文标题
熵有效的饰面编码
Entropy-efficient finitary codings
论文作者
论文摘要
我们表明，I.I.D过程的任何有限的，可数值的可计值的限制因子也可以表示为有限价值的I.I.D过程的熵，其熵是任意接近目标过程的。作为一个应用程序，我们对van den Berg和Steif的问题给出了肯定的答案，内容涉及$ \ Mathbb {z}^d $上的关键ISING模型。在途中，我们证明了有关限量异构和限制因素的几个结果。我们的结果是在一个新的框架中开发的，以使一个可计数集满足特定属性的置换组不变。这个新的框架包括所有可数符合的群体上的所有``经典''过程，以及具有``唯一中心的球''上的及时可及格图上的所有不变过程。我们的一些结果已经是$ \ Mathbb {z} $ - 流程的新结果。我们证明了Smorodinsky的同构定理的相对版本，用于有限依赖的$ \ mathbb {z} $ - processes。我们还将Keane--Smorodinsky的同构定理扩展到可数值的I.I.D过程，并将其在波兰空间中的值延伸到I.I.D过程。
We show that any finite-entropy, countable-valued finitary factor of an i.i.d process can also be expressed as a finitary factor of a finite-valued i.i.d process whose entropy is arbitrarily close to the target process. As an application, we give an affirmative answer to a question of van den Berg and Steif about the critical Ising model on $\mathbb{Z}^d$. En route, we prove several results about finitary isomorphisms and finitary factors. Our results are developed in a new framework for processes invariant to a permutation group of a countable set satisfying specific properties. This new framework includes all ``classical'' processes over countable amenable groups and all invariant processes on transitive amenable graphs with ``uniquely centered balls''. Some of our results are new already for $\mathbb{Z}$-processes. We prove a relative version of Smorodinsky's isomorphism theorem for finitely dependent $\mathbb{Z}$-processes. We also extend the Keane--Smorodinsky finitary isomorphism theorem to countable-valued i.i.d processes and to i.i.d processes taking values in a Polish space.
