  This paper introduces heat semigroups of topological Markov chains and
Cuntz-Krieger algebras by means of spectral noncommutative geometry. Using
recent advances on the logarithmic Dirichlet Laplacian on Ahlfors regular
metric-measure spaces, we construct spectral triples on Cuntz-Krieger algebras
from singular integral operators. These spectral triples exhaust K-homology and
for Cuntz algebras we can compute their heat operators explicitly as Riesz
potential operators. We also describe their isometry group in terms of the
automorphism group of the underlying directed graph and prove that the
Voiculescu noncommutative topological entropy vanishes on isometries.
