  The first aim of this paper is to introduce a class of Markov chains on
$\mathbb{Z}_+$ which are discrete self-similar in the sense that their
semigroups satisfy an invariance property expressed in terms of a discrete
random dilation operator. After showing that this latter property requires the
chains to be upward skip-free, we first establish a gateway relation, a concept
introduced in [26], between the semigroup of such chains and the one of
spectrally negative self-similar Markov processes on $\mathbb{R}_+$. As a
by-product, we prove that each of these Markov chains, after an appropriate
scaling, converge in the Skorohod metric, to the associated self-similar Markov
process. By a linear perturbation of the generator of these Markov chains, we
obtain a class of ergodic Markov chains, which are non-reversible. By means of
intertwining and interweaving relations, where the latter was recently
introduced in [27], we derive several deep analytical properties of such
ergodic chains including the description of the spectrum, the spectral
expansion of their semigroups, the study of their convergence to equilibrium in
the $\Phi$-entropy sense as well as their hypercontractivity property.
