  The dynamical correlations of a strongly correlated system is an essential
ingredient to describe its non-equilibrium properties. We present a general
method to calculate exactly the dynamical correlations of hard-core anyons in
one-dimensional lattices, valid for any type of confining potential and any
temperature. We obtain exact explicit expressions of the Green's function, the
spectral function, and the out-of-time-ordered correlators (OTOCs). We find
that the anyonic spectral function displays three main singularity lines which
can be explained as a double spectrum in analogy to the Lieb-Liniger gas. The
dispersion relations of these lines can be given explicitly and they cross at a
\emph{hot point} $(q_m,\omega_m)$, which induces a peak in the momentum
distribution function at $q_m$ and a power-law singularity in the local
spectral function at $\omega_m$. We also find that the anyonic statistics can
induces spatial asymmetry in the Green's function, its spectrum, and the OTOC.
Moreover, the information spreading characterized by the OTOCs shows light-cone
dynamics, asymmetric for general statistics and low temperatures, but symmetric
at infinite temperature. Our results pave the way toward studying the
non-equilibrium dynamics of hard-core anyons and experimentally probing anyonic
statistics through spectral functions.
