We study a one-parameter family of countably piecewise linear interval maps, which, although Markov, fail the `large image property'. This leads to conservative as well as dissipative behaviour for different maps in the family with respect to Lebesgue. We investigate the transition between these two types, and study the associated thermodynamic formalism, describing in detail the second order phase transitions (i.e. the pressure function is $C^1$ but not $C^2$ at the phase transition) that occur in transition to dissipativity. We also study the various natural definitions of pressure which arise here, computing these using elementary recurrence relations
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