Guido AHUMADA (IRIMAS, Université de Haute-Alsace)
Given an increasing odd homeomorphism σ : R → R, the two bijective maps hσ , vσ : R2 → R2 defined by
hσ(x, y) = (x + σ-1(y), y)
vσ(x, y) = (x, σ(x) + y)
are called generalized transvections. We study the action on the plane of the group Γ(σ) generated by these two maps. Particularly interesting cases arise when σ(x) = sgn(x)|x|α. We prove that most points have dense orbits and that every nonzero point has a dense orbit when σ(x) = sgn(x)|x|2. We also look at invariant measures and thanks to Nogueira’s work about SL(2, Z)-invariant measure, we can determine these measures when σ is linear in a neighborhood of the origin.
Joint work with Nicolas Chevallier.
I2M - Château-Gombert
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Andrey ZVYAGIN (Voronezh State University, Russian Federation)
The initial–boundary value problems under consideration describe the weakly concentrated water polymer solutions motion. In particular, Voigt model, Kelvin–Voigt model, the second grade order model will be considered. In this mathematical models the viscosity depends on a temperature, which leads to the emergence of additional heat balance equation (it is a parabolic equation with nonsmooth coefficients and with the right part from L1(0,T ;L1(Ω)). For these initial–boundary value problems under consideration the existence theorems of weak solutions will be proved. For this the topological approximation approach for hydrodynamic problems is used.