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Register a new userOn Discrete Subgroups of automorphism of $P^2_C$
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We study the geometry and dynamics of discrete subgroups $Gamma$ of $PSL(3,mathbb{C})$ with an open invariant set $Omega subset PC^2$ where the action is properly discontinuous and the quotient $Omega/Gamma$ contains a connected component whicis compact. We call such groups {it quasi-cocompact}. In this case $Omega/Gamma$ is a compact complex projective orbifold and $Omega$ is a {it divisible set}. Our first theorem refines classical work by Kobayashi-Ochiai and others about complex surfaces with a projective structure: We prove that every such group is either virtually affine or complex hyperbolic. We then classify the divisible sets that appear in this way, the corresponding quasi-cocompact groups and the orbifolds $Omega/Gamma$. We also prove that excluding a few exceptional cases, the Kulkarni region of discontinuity coincides with the equicontinuity region and is the largest open invariant set where the action is properly discontinuous.
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Let $ G $ be a discrete subgroup of PU(1,n). Then $ G $ acts on $mathbb {P}^n_mathbb C$ preserving the unit ball $mathbb {H}^n_mathbb {C}$, where it acts by isometries with respect to the Bergman metric. In this work we determine the equicontinuty region $Eq(G)$ of $G$ in $mathbb P^n_{mathbb C}$: It is the complement of the union of all complex projective hyperplanes in $mathbb {P}^n_{mathbb C}$ which are tangent to $partial mathbb {H}^n_mathbb {C}$ at points in the Chen-Greenberg limit set $Lambda_{CG}(G )$, a closed $G$-invariant subset of $partial mathbb {H}^n_mathbb {C}$, which is minimal for non-elementary groups. We also prove that the action on $Eq(G)$ is discontinuous.
In this article we study automorphisms of Toeplitz subshifts. Such groups are abelian and any finitely generated torsion subgroup is finite and cyclic. When the complexity is non superlinear, we prove that the automorphism group is, modulo a finite cyclic group, generated by a unique root of the shift. In the subquadratic complexity case, we show that the automorphism group modulo the torsion is generated by the roots of the shift map and that the result of the non superlinear case is optimal. Namely, for any $varepsilon > 0$ we construct examples of minimal Toeplitz subshifts with complexity bounded by $C n^{1+epsilon}$ whose automorphism groups are not finitely generated. Finally, we observe the coalescence and the automorphism group give no restriction on the complexity since we provide a family of coalescent Toeplitz subshifts with positive entropy such that their automorphism groups are arbitrary finitely generated infinite abelian groups with cyclic torsion subgroup (eventually restricted to powers of the shift).
Let $M$ be a triangulable compact manifold. We prove that, among closed subgroups of $homeo_{0}(M)$ (the identity component of the group of homeomorphisms of $M$), the subgroup consisting of volume preserving elements is maximal.
We completely describe the weak closure of the powers of the Koopman operator associated to Chacons classical automorphism. We show that weak limits of these powers are the ortho-projector to constants and an explicit family of polynomials. As a consequence, we answer negatively the question of alpha-weak mixing for Chacons automorphism.
It has been recently proved that the automorphism group of a minimal subshift with non-superlinear word complexity is virtually $mathbb{Z}$ [DDPM15, CK15]. In this article we extend this result to a broader class proving that the automorphism group of a minimal S-adic subshift of finite alphabet rank is virtually $mathbb{Z}$. The proof is based on a fine combinatorial analysis of the asymptotic classes in this type of subshifts, which we prove are a finite number.
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