— Let r > 1 be a real. In this paper, we study piecewise class P Cr circle homeomorphisms with irrational rotation numbers. We give characterizations for such homeomorphisms that are piecewise Cr conjugate to Cr diffeomorphisms. As a consequence, we obtain a criterion of piecewise Cr conjugacy to diophantine rotations. This characterization extends those obtained by Liousse for the PL circle ho...

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Given any positive sequence {cn}n∈N, we construct orientation preserving homeomorphisms f : R → R such that Fix(f) = Per(f) = {0}, 0 is Lyapunov stable and lim sup |i(f ,0)| cm = ∞. We will use our results to discuss and to point out some strong di erences with respect to the computation and behavior of the sequences of the indices of planar homeomorphisms.

We show that the group of piecewise-linear homeomorphisms of R having bounded slopes surjects onto the group QI(R) of all quasi-isometries of R. We prove that the following groups can be imbedded in QI(R): the group of compactly supported piecewise-linear homeomorphisms of R, the Richard Thompson group F , and the free group of continuous rank.

We introduce and characterize extremal 2-cuts for good fractal necklaces. Using this characterization the related topological properties of 2-cuts, we prove that every necklace has a unique IFS in certain sense. Also, two necklaces admit only rigid homeomorphisms thus group self-homeomorphisms is countable. In addition, weaker co-Hopfian property also obtained.