8 research outputs found
Asymptotic pressure on Cayley graphs of finitely generated semigroups
The vertices of the Cayley graph of a finitely generated semigroup form a set
of sites which can be labeled by elements of a finite alphabet in a manner
governed by a nonnegative real interaction matrix, respecting nearest neighbor
adjacency restrictions. To the set of these configurations one can associate a
pressure, which is defined as the limit, when it exists, of averages of the
logarithm of the partition function over certain finite subgraphs. We prove
that for shifts of finite type on generalized Fibonacci trees, under an added
condition, the limit exists and is given by an infinite series. We also show
that the limit of any cluster points of the pressure on finite subtrees as the
number of generators grows without bound, which we call the asymptotic
pressure, equals the logarithm of the maximum row sum of the interaction
matrix.Comment: Clarified a hypothesis of the first theorem and the proof of the
secon
Topological Entropy for Shifts of Finite Type Over and Tree
We study the topological entropy of hom tree-shifts and show that, although
the topological entropy is not conjugacy invariant for tree-shifts in general,
it remains invariant for hom tree higher block shifts. In
doi:10.1016/j.tcs.2018.05.034 and doi:10.3934/dcds.2020186, Petersen and Salama
demonstrated the existence of topological entropy for tree-shifts and
, where is the hom tree-shift
derived from . We characterize a necessary and sufficient condition when the
equality holds for the case where is a shift of finite type. In addition,
two novel phenomena have been revealed for tree-shifts. There is a gap in the
set of topological entropy of hom tree-shifts of finite type, which makes such
a set not dense. Last but not least, the topological entropy of a reducible hom
tree-shift of finite type is equal to or larger than that of its maximal
irreducible component
The strip entropy approximation of Markov shifts on trees
The strip entropy is studied in this article. We prove that the strip entropy
approximation is valid for every ray of a golden-mean tree. This result extends
the previous result of [Petersen-Salama, Discrete \& Continuous Dynamical
Systems, 2020] on the conventional 2-tree. Lastly, we prove that the strip
entropy approximation is valid for eventually periodic rays of a class of
Markov-Cayley trees
On the topological pressure of axial product on trees
This article investigates the topological pressure of isotropic axial
products of Markov subshift on the -tree. We show that the quantity
increases with dimension , and demonstrate that, with the introduction of
surface pressure, the two types of pressure admit the same asymptotic value. To
this end, the pattern distribution vectors and the associated transition
matrices are introduced herein to partially transplant the large deviation
theory to tree-shifts, and so the increasing property is proved via an almost
standard argument. An application of the main result to a wider class of shift
spaces is also provided in this paper, and numerical experiments are included
for the purpose of verification