Tree-shifts: the entropy of tree-shifts of finite type*This work is partially supported by the Ministry of Science and Technology, ROC (Contract No MOST 105-2115-M-259 -006 -MY2 and 105-2115-M-390 -001 -MY2). The first author is partially supported by National Center for Theoretical Sciences. (1st June 2017)

• Record Type: Journal Article
• Authors: Ban, Jung-Chao; Chang, Chih-Hung
• Is Part Of: Nonlinearity. Volume 30:Number 7(2017:Jul.)
• Page Start: 2785
• Page End: 2804
• Publication Date: 2017-06-01
• Subjects: tree-shift of finite type -- entropy -- hidden entropy; 37B10; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/aa72c0
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: This paper studies the entropy of tree-shifts of finite type with and without boundary conditions. We demonstrate that computing the entropy of a tree-shift of finite type is equivalent to solving a system of nonlinear recurrence equations. Furthermore, the entropy of the binary Markov tree-shifts over two symbols is either 0 or ln ⁡ 2 . Meanwhile, the realization of a class of reals including multinacci numbers is elaborated, which indicates that tree-shifts are capable of rich phenomena. By considering the influence of three different types of boundary conditions, say, the periodic, Dirichlet, and Neumann boundary conditions, the necessary and sufficient conditions for the coincidence of entropy with and without boundary conditions are addressed.