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We investigate the problem of when big mapping class groups are generated by involutions. Restricting our attention to the class of self-similar surfaces, which are surfaces with self-similar ends space, as defined by Mann and Rafi, and with 0 or infinite genus, we show that, when the set of maximal ends is infinite, then the mapping class groups of these surfaces are generated by involutions, normally generated by a single involution, and uniformly perfect. In fact, we derive this statement as a corollary of the corresponding statement for the homeomorphism groups of these surfaces. On the other hand, among self-similar surfaces with one maximal end, we produce infinitely many examples in which their big mapping class groups are neither perfect nor generated by torsion elements. These groups also do not have the automatic continuity property.
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Using the existence of certain symplectic submanifolds in symplectic 4-manifolds, we prove an estimate from above for the number of singular fibers with separating vanishing cycles in minimal Lefschetz fibrations over surfaces of positive genus. This estimate is then used to deduce that mapping class groups are not uniformly perfect, and that the map from their second bounded cohomology to ordinary cohomology is not injective.
We provide explicit, simple, geometric formulas for free involutions rho of Euclidean spheres that are not conjugate to the antipodal involution. Therefore the quotient S^n/rho is a manifold that is homotopically equivalent but not diffeomorphic to RP^n. We use these formulas for constructing explicit non-trivial elements in pi_1 Diff(S^5) and pi_1 Diff(S^13) and to provide explicit formulas for non-cancellation phenomena in group actions.
The article contains a construction of a self-similar dendryte which cannot be the attractor of any self-similar zipper.
Self-similarity in the network traffic has been studied from several aspects: both at the user side and at the network side there are many sources of the long range dependence. Recently some dynamical origins are also identified: the TCP adaptive congestion avoidance algorithm itself can produce chaotic and long range dependent throughput behavior, if the loss rate is very high. In this paper we show that there is a close connection between the static and dynamic origins of self-similarity: parallel TCPs can generate the self-similarity themselves, they can introduce heavily fluctuations into the background traffic and produce high effective loss rate causing a long range dependent TCP flow, however, the dropped packet ratio is low.
The problem of reconstructing functions from their asymptotic expansions in powers of a small variable is addressed by deriving a novel type of approximants. The derivation is based on the self-similar approximation theory, which presents the passage from one approximant to another as the motion realized by a dynamical system with the property of group self-similarity. The derived approximants, because of their form, are named the self-similar factor approximants. These complement the obtained earlier self-similar exponential approximants and self-similar root approximants. The specific feature of the self-similar factor approximants is that their control functions, providing convergence of the computational algorithm, are completely defined from the accuracy-through-order conditions. These approximants contain the Pade approximants as a particular case, and in some limit they can be reduced to the self-similar exponential approximants previously introduced by two of us. It is proved that the self-similar factor approximants are able to reproduce exactly a wide class of functions which include a variety of transcendental functions. For other functions, not pertaining to this exactly reproducible class, the factor approximants provide very accurate approximations, whose accuracy surpasses significantly that of the most accurate Pade approximants. This is illustrated by a number of examples showing the generality and accuracy of the factor approximants even when conventional techniques meet serious difficulties.
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