Using quantitative perturbation theory for linear operators, we prove spectral gap for transfer operators of various families of intermittent maps with almost constant potentials (high-temperature regime). Holder and bounded p-variation potentials are treated, in each case under a suitable assumption on the map, but the method should apply more generally. It is notably proved that for any Pommeau-Manneville map, any potential with Lispchitz constant less than 0.0014 has a transfer operator acting on Lip([0, 1]) with a spectral gap; and that for any 2-to-1 unimodal map, any potential with total variation less than 0.0069 has a transfer operator acting on BV([0, 1]) with a spectral gap. We also prove under quite general hypotheses that the classical definition of spectral gap coincides with the formally stronger one used in (Giulietti et al. 2015), allowing all results there to be applied under the high temperature bounds proved here: analyticity of pressure and equilibrium states, central limit theorem, etc.
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