Abstract: We consider piecewise cone-hyperbolic systems satisfying a bunching condition, and we obtain a bound on the essential spectral radius of the associated weighted transfer operators acting on anisotropic Sobolev spaces. The bunching condition is always satisfied in dimension two, and our results give a unifying treatment of the work of Demers-Liverani  and our previous work . When the complexity is subexponential, our bound implies a spectral gap for the transfer operator corresponding to the physical measures in many cases (for example if $T$ preserves volume, or if the stable dimension is equal to $1$ and the unstable dimension is not zero).
Keywords: hyperbolic systems with singularities, transfer operators, spectral gap,
anisotropic Sobolev spaces, physical/SRB measures, exponential decay of correlations, piecewise
Received: July 2009; Revised: March 2010; Published: May 2010.