Hyperbolicity of renormalization for maps with multiple breaks

We construct hyperbolic horseshoes for piecewise-analytic homeomorphisms of the circle with {\it multiple} break-type singularities, under the assumptions of bounded type rotation numbers and bounded geometry – provided the sizes of the breaks are uniformly small. As a consequence, we obtain a $C^{1+\alpha }$-rigidity result for such maps and prove that rigidity classes are analytic submanifolds.