一类$\zz^2$马尔可夫随机场的逼近熵和一类函数在有限型$\zz^2$移位上的压力
Approximating entropy for a class of $\zz^2$ Markov Random Fields and pressure for a class of functions on $\zz^2$ shifts of finite type
摘要：
对于一类$\zz^2$马尔可夫随机场（MRF）$μ$，我们证明了在高度为$n$的条带上的诱导MRF的熵的连续差序列指数快速地（在$n$中）收敛到$μ$的熵.当$μ$是由有限型$X$的强不可约最近邻$\zz^2$位移上的最近邻相互作用给出的吉布斯态时，这些条熵可以显式计算。我们的状态这一结果的近似（拓扑）压力的某些功能，这样的$X$，我们表明，这些压力是可计算的，如果值所采取的功能是可计算的。最后，我们表明，我们的结果适用于硬核模型和伊辛模型的某些参数值的相应的相互作用，以及某些最近邻的有限型的$\zz^2$移位的拓扑熵，推广的结果在\cite{Pa}。
英文摘要：
For a class of $\zz^2$ Markov Random Fields (MRFs) $μ$, we show that the sequence of successive differences of entropies of induced MRFs on strips of height $n$ converges exponentially fast (in $n$) to the entropy of $μ$. These strip entropies can be computed explicitly when $μ$ is a Gibbs state given by a nearest-neighbor interaction on a strongly irreducible nearest-neighbor $\zz^2$ shift of finite type $X$. We state this result in terms of approximations to the (topological) pressures of certain functions on such an $X$, and we show that these pressures are computable if the values taken on by the functions are computable. Finally, we show that our results apply to the hard core model and Ising model for certain parameter values of the corresponding interactions, as well as to the topological entropy of certain nearest-neighbor $\zz^2$ shifts of finite type, generalizing a result in \cite{Pa}.
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