  (abridged) We present a detailed reanalysis of the Chandra data for the
galaxy cluster Abell 4059 and its central radio galaxy, PKS2354-35. We also
present new 1.4GHz and 4.7GHz CnB-array radio data from the Very Large Array
(VLA), as well as a short archival WFPC2 image from the Hubble Space Telescope.
The presence of a strong interaction between this radio galaxy and the
intracluster medium (ICM) was suggested by Huang & Sarazin (1998) on the basis
of a short observation by the High Resolution Imager on ROSAT, and confirmed in
our preliminary analysis of the Chandra/ACIS-S data. In particular, X-ray
imaging clearly shows two cavities within the ICM that are approximately
aligned with the radio-galaxy axis. However, using our new radio maps we fail
to find a detailed correspondence between the 1 arcmin scale radio-lobes and
the ICM cavities. This suggests that the cavities are ``ghosts'' of a previous
burst of powerful activity by PKS 2354-35. We also examine the nature of the
central asymmetric ridge (or bar) of X-ray emission extending for 30kpc
south-west of the cluster center that has been noted in these previous
analyzes. We find the ridge to be denser and cooler than, but probably in
pressure balance with, its surroundings. The thermal evolution of this
structure seems to be dominated by radiative cooling, possibly enhanced by the
radio-galaxy ICM interaction. We discuss several possible models for the
formation of this SW ridge and find none of them to be entirely satisfactory.
