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Register a new userThe limit set for discrete complex hyperbolic groups
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Given a discrete subgroup $Gamma$ of $PU(1,n)$ it acts by isometries on the unit complex ball $Bbb{H}^n_{Bbb{C}}$, in this setting a lot of work has been done in order to understand the action of the group. However when we look at the action of $Gamma$ on all of $ Bbb{P}^n_{Bbb{C}}$ little or nothing is known, in this paper study the action in the whole projective space and we are able to show that its equicontinuity agree with its Kulkarni discontuity set. Morever, in the non-elementary case, this set turns out to be the largest open set on which the group acts properly and discontinuously and can be described as the complement of the union of all complex projective hyperplanes in $ Bbb{P}^n_{Bbb{C}}$ which are tangent to $partial Bbb{H}^n_{Bbb{C}}$ at points in the Chen-Greenberg limit set $Lambda_{CG}(Gamma)$.
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Let $Gamma$ be a torsion-free hyperbolic group. We study $Gamma$--limit groups which, unlike the fundamental case in which $Gamma$ is free, may not be finitely presentable or geometrically tractable. We define model $Gamma$--limit groups, which always have good geometric properties (in particular, they are always relatively hyperbolic). Given a strict resolution of an arbitrary $Gamma$--limit group $L$, we canonically construct a strict resolution of a model $Gamma$--limit group, which encodes all homomorphisms $Lto Gamma$ that factor through the given resolution. We propose this as the correct framework in which to study $Gamma$--limit groups algorithmically. We enumerate all $Gamma$--limit groups in this framework.
We study the dynamics of towers defined by fixed points of renormalization for Feigenbaum polynomials in the complex plane with varying order ell of the critical point. It is known that the measure of the Julia set of the Feigenbaum polynomial is positive if and only if almost every point tends to 0 under the dynamics of the tower for corresponding ell. That in turn depends on the sign of a quantity called the drift. We prove the existence and key properties of absolutely continuous invariant measures for tower dynamics as well as their convergence when ell tends to infinity. We also prove the convergence of the drifts to a finite limit which can be expressed purely in terms of the limiting tower which corresponds to a Feigenbaum map with a flat critical point
We prove that a hyperbolic Dulac germ with complex coefficients in its expansion is linearizable on a standard quadratic domain and that the linearizing coordinate is again a complex Dulac germ. The proof uses results about normal forms of hyperbolic transseries from another work of the authors.
The well-known plastic number substitution gives rise to a ternary inflation tiling of the real line whose inflation factor is the smallest Pisot-Vijayaraghavan number. The corresponding dynamical system has pure point spectrum, and the associated control point sets can be described as regular model sets whose windows in two-dimensional internal space are Rauzy fractals with a complicated structure. Here, we calculate the resulting pure point diffraction measure via a Fourier matrix cocycle, which admits a closed formula for the Fourier transform of the Rauzy fractals, via a rapidly converging infinite product.
In this note, we show that the exceptional algebraic set of an infinite discrete group in $PSL(3,Bbb{C})$ should be a finite union of complex lines, copies of the Veronese curve or copies of the cubic $xy^2-z^3$.
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