This paper first studies the regularity of conformal homeomorphisms on smooth locally embeddable strongly pseudoconvex CR manifolds. Then moduli of curve families are used to estimate the maximal dilatations of quasiconformal homeomorphisms. On certain CR 3-manifolds, namely, CR circle bundles over flat tori, extremal quasiconformal homeomorphisms in some homotopy classes are constructed. These...

We call a dynamical system on a measurable metric space measureexpansive if the probability of two orbits remain close each other for all time is negligible (i.e. zero). We extend results of expansive systems on compact metric spaces to the measure-expansive context. For instance, the measureexpansive homeomorphisms are characterized as those homeomorphisms f for which the diagonal is almost in...

In this paper we describe the notion of a weak lipschitzianity of a mapping on a C stratification. We also distinguish a class of regularity conditions that are in some sense invariant under definable, locally Lipschitz and weakly bi-Lipschitz homeomorphisms. This class includes the Whitney (B) condition and the Verdier condition.

We compute the generalized Lefschetz number of orientation-preserving selfhomeomorphisms of a compact punctured disk, using the fact that homotopy classes of these homeomorphisms can be identified with braids. This result is applied to study Nielsen-Thurston canonical homeomorphisms on a punctured disk. We determine, for a certain class of braids, the rotation number of the corresponding canoni...

Let M be a compact manifold, possibly with boundary. We show that the group of homeomorphisms of M has the automatic continuity property : any homomorphism from Homeo(M) to any separable group is necessarily continuous. This answers a question of C. Rosendal. If N ⊂ M is a submanifold, the group of homeomorphisms of M that preserve N also has this property. Various applications of automatic con...

Let B n be the braid group on n 4 strands. We prove that B n modulo its center is co-Hoppan. We then show that any injective endomorphism of B n is geometric in the sense that it is induced by a homeomorphism of a punctured disk. We further prove that any injection from B n to B n+1 is geometric for n 7. Additionally, we obtain analogous results for mapping class groups of punctured spheres. Th...

On subsets of the Mandelbrot set, EM ⊂M, homeomorphisms are constructed by quasi-conformal surgery. When the dynamics of quadratic polynomials is changed piecewise by a combinatorial construction, a general theorem yields the corresponding homeomorphism h : EM → EM in the parameter plane. Each h has two fixed points in EM , and a countable family of mutually homeomorphic fundamental domains. Po...