  We aim at an explicit characterization of the renormalized Hamiltonian after
decimation transformation of a one-dimensional Ising-type Hamiltonian with a
nearest-neighbor interaction and a magnetic field term. To facilitate a deeper
understanding of the decimation effect, we translate the renormalization flow
on the Ising Hamiltonian into a flow on the associated Markov chains through
the Markov-Gibbs equivalence. Two different methods are used to verify the
well-known conjecture that the eigenvalues of the linearization of this
renormalization transformation about the fixed point bear important information
about all six of the critical exponents. This illustrates the universality
property of the renormalization group map in this case.
