On the Renormalizations of Circle Homeomorphisms with Several Break Points

Hausdorff Dimension of Invariant Measures of Multicritical Circle Maps

“…Let us point out that the aforementioned smooth linearization results in [3] rely heavily on the results of [13] concerning the behavior of subsequent renormalizations of circle maps without periodic points and zero mean nonlinearity. Let us mention that the behaviour on the renormalization of piecewise smooth circle maps was previously known only for transformations with bounded combinatorics [5], [7]. We refer the interested reader to [5] or [7] for the definition of bounded combinatorics.…”

“…Let us mention that the behaviour on the renormalization of piecewise smooth circle maps was previously known only for transformations with bounded combinatorics [5], [7]. We refer the interested reader to [5] or [7] for the definition of bounded combinatorics.…”

On the Hausdorff dimension of invariant measures of piecewise smooth circle homeomorphisms

“…In fact, when seen as a circle map, the successive renormalizations of a P-homeomorphism appear as a subsequence of the successive renormalizations of the same map when considered a GIET. Let us mention that this approach has been successfully used to study renormalizations and rigidity properties for an exceptional class of P-homeomorphisms in [6][7][8], and to study rigidity properties of P-homeomorphisms with zero mean nonlinearity in [3].…”