Let $ \mathcal{A} $ be an alphabet, the full $\mathcal{A}-shift$ is the collection of all bi-infite sequences with symbols of $ \mathcal{A} $. The full $ \mathcal{A}-shift $ is denoted by
\[\mathcal{A}^{\mathbb{Z}}=\{ (x_i)_{i \in \mathbb{Z}} : x_i \in \mathcal{A} , \ for \ all \ i \in \mathbb{Z} \} \]

The shift map is the map $\sigma:\mathcal{A}^{\mathbb{Z}}\to\mathcal{A}^{\mathbb{Z}}$ which associates each point $ x \in \mathcal{A}^\mathbb{Z} $ to the point $\sigma(x) $ whose $ i^{th}$ coordinate is $x_{i+1} $.

A subset $ \mathcal{S} \subseteq \mathcal{A}^{\mathbb{Z}}$ is called a subshift over $ \mathcal{A} $ if $ \mathcal{S} $ is closed with respect to the topology of $ \mathcal{A}^{\mathbb{Z}} $ and if $ \mathcal{S} $ is invariant under the shift map, that is, $ \sigma ( \mathcal{S} ) \subseteq \mathcal{S} $.

A Markov shift is a subshift which can be associated to a set $ \mathcal{S}_G $ of bi-infinite walks on the edges of a countable directed graph $ G $. The Markov shift $ \mathcal{S} $ is locally compact if and only if $ G $ has finite in- and out-degree. The Markov shift is compact if and only if $ G $ is a finite graph if and only if it is a shift finite.

For a Markov shift $ \mathcal{S} $, let $ Per ( \mathcal{S} ) $ the set of periodic points of $ \mathcal{S} $, that is, $ Per ( \mathcal{S} ) = \{ y \in \mathcal{S} : \mathcal{S}^n (y) = y \ for \ any \ n \in \mathbb{N} \} $.

Let $ \mathcal{S} $, $ \mathcal{T} $ subshifts, a factor map $ f : \mathcal{S} \rightarrow \mathcal{T} $ is a continuous shift commuting onto map. We say that a fiber of $ f $ on $ y \in \mathcal{T} $ is the preimage set $ f^{-1} (y) $. Moreover, $ f $ is said to be bounded-to-1 if there is some $ M \in \mathbb{N} $ such that all fibers of $ f $ have cardinality at most $ M $; $ f $ is finite-to-1 if all fibers are finite sets; and $ f $ is countable-to-1 if all fibers are countable sets.

The 1-point compactification $ \mathcal{\mathcal{S}_{0}}$ of a locally compact subshift $ \mathcal{S} $ is the compact metric dynamical system which consists of the Alexandroff 1-point compactification of the shift space with the extended shift maps.

The Gurevic entropy is defined to be the topological entropy of the 1-point compactification of the subshift.
\[h_{G}(\mathcal{S})=h_{top}(\mathcal{S}_{0})\]

We consider the Gurevic metric the metric on $ \mathcal{S} $ such that the completion of $ \mathcal{S} $ with respect to this metric is $ \mathcal{S}_{0} $.

Let $ \mathcal{S} $, $ \mathcal{T} $ transitive locally compact Markov shifts. A factor map $ f : \mathcal{S} \rightarrow \mathcal{T} $ is proper if $ f^{-1}(K) $ is a compact set for every compact set $ K \subseteq \mathcal{T} $.

In this work, we present relationships between the Gurevic entropies of two transitive locally compact Markov shifts under some conditions on the factor maps between them.

\textbf{Theorem:} Let $ \mathcal{S} $ be a transitive locally compact Markov shift and $ \mathcal{T} $ a subshift locally compact. Let $ f : \mathcal{S} \rightarrow \mathcal{T} $ a factor map. If the fiber $ f^{-1} (y) $ is countable for every $ y \in Per ( \mathcal{T} ) $ then
\[ h_{G}(\mathcal{S}) \leq h_{G}(\mathcal{T}). \]
In particular, if $ f $ is countable-to-1, so $ h_{G}(\mathcal{S}) \leq h_{G}(\mathcal{T}) $.

Theorem: Let $ \mathcal{S} , \mathcal{T} $ be transitive locally compact Markov shifts and let $ f : \mathcal{S} \rightarrow \mathcal{T} $ a factor map finite-to-1. Then \[ h_{G}(\mathcal{S}) = h_{G}(\mathcal{T}). \]

Theorem: Let $ \mathcal{S} , \mathcal{T} $ be locally compact Markov subshifts and let $ f : \mathcal{S} \rightarrow \mathcal{T} $ a factor map proper. Then \[ h_{G}(\mathcal{S}) \geq h_{G}(\mathcal{T}). \]

Theorem: Let $ \mathcal{S} , \mathcal{T} $ be locally compact Markov subshifts and let $ f : \mathcal{S} \rightarrow \mathcal{T} $ a factor map countable-to-1 proper. Then \[ h_{G}(\mathcal{S}) = h_{G}(\mathcal{T}). \]

REFERENCES

Doris Fiebig: Factor maps, entropy and fiber cardinality for Markov shifts. Rocky Mountain Journal of Mathematics, Pages 955 - 986, Volume 31, Number 3, Fall 2001.

Douglas Lind, Brian Marcus: An introduction to symbolic dynamics and coding. Cambridge University Press, 1995.

Bruce Kitchens: Symbolic Dynamics: One-sided, Two-sided and Countable State Markov Shifts. Springer Verlag, 1997.