scholarly journals Mixing Flows on Moduli Spaces of Flat Bundles over Surfaces

2018 ◽  
pp. 519-534
Author(s):  
Giovanni Forni ◽  
William M. Goldman
Keyword(s):  
Moduli Spaces ◽  
Mapping Class ◽  
Deformation Space ◽  
Continuous Version ◽  
Local Flow ◽  
Deformation Spaces ◽  
Ergodic Properties ◽  
Strongly Mixing ◽  
Flat Bundle ◽  
Teichmuller Dynamics

This chapter extends Teichmüller dynamics to a flow on the total space of a flat bundle of deformation spaces of representations of the fundamental group of a fixed surface S in a Lie group G. The resulting dynamical system is a continuous version of the action of the mapping class group of S on the deformation space. It observes how ergodic properties of this action relate to this flow. When G is compact, this flow is strongly mixing over each component of the deformation space and of each stratum of the Teichmüller unit sphere bundle over the Riemann moduli space. It proves ergodicity for the analogous lift of the Weil–Petersson geodesic local flow.

scholarly journals Picard Groups of Moduli Spaces of Curves with Symmetry

2018 ◽  
Vol 2020 (23) ◽  
pp. 9293-9335
Author(s):  
Kevin Kordek
Keyword(s):  
Moduli Spaces ◽  
Topological Type ◽  
Hyperelliptic Curves ◽  
Mapping Class ◽  
Finitely Generated ◽  
Smooth Curves ◽  
Group Of Automorphisms ◽  
Picard Groups ◽  
Moduli Spaces Of Curves ◽  
Special Cases

Abstract We study the Picard groups of moduli spaces of smooth complex projective curves that have a group of automorphisms with a prescribed topological action. One of our main tools is the theory of symmetric mapping class groups. In the 1st part of the paper, we show that, under mild restrictions, the moduli spaces of smooth curves with an abelian group of automorphisms of a fixed topological type have finitely generated Picard groups. In certain special cases, we are able to compute them exactly. In the 2nd part of the paper, we show that finite abelian level covers of the hyperelliptic locus in the moduli space of smooth curves have finitely generated Picard groups. We also compute the Picard groups of the moduli spaces of hyperelliptic curves of compact type.

scholarly journals Deformation spaces and normal forms around transversals

2020 ◽  
Vol 156 (4) ◽  
pp. 697-732 ◽  
Author(s):  
Francis Bischoff ◽  
Henrique Bursztyn ◽  
Hudson Lima ◽  
Eckhard Meinrenken
Keyword(s):  
Normal Form ◽  
Normal Bundle ◽  
Normal Forms ◽  
Model Structure ◽  
Deformation Space ◽  
Local Behavior ◽  
Courant Algebroids ◽  
Deformation Spaces ◽  
Singular Foliations ◽  
Normal Form Theorem

Given a manifold $M$ with a submanifold $N$, the deformation space ${\mathcal{D}}(M,N)$ is a manifold with a submersion to $\mathbb{R}$ whose zero fiber is the normal bundle $\unicode[STIX]{x1D708}(M,N)$, and all other fibers are equal to $M$. This article uses deformation spaces to study the local behavior of various geometric structures associated with singular foliations, with $N$ a submanifold transverse to the foliation. New examples include $L_{\infty }$-algebroids, Courant algebroids, and Lie bialgebroids. In each case, we obtain a normal form theorem around $N$, in terms of a model structure over $\unicode[STIX]{x1D708}(M,N)$.

scholarly journals The minimally displaced set of an irreducible automorphism is locally finite

2020 ◽  
Vol 55 (2) ◽  
pp. 301-336
Author(s):  
Stefano Francaviglia ◽  
◽  
Armando Martino ◽  
Dionysios Syrigos ◽  
◽  
...  
Keyword(s):  
Free Group ◽  
Outer Space ◽  
General Setting ◽  
Deformation Space ◽  
Train Track ◽  
Free Products ◽  
Deformation Spaces ◽  
Locally Finite ◽  
Irreducible Automorphism

We prove that the minimally displaced set of a relatively irreducible automorphism of a free splitting, situated in a deformation space, is uniformly locally finite. The minimally displaced set coincides with the train track points for an irreducible automorphism. We develop the theory in a general setting of deformation spaces of free products, having in mind the study of the action of reducible automorphisms of a free group on the simplicial bordification of Outer Space. For instance, a reducible automorphism will have invariant free factors, act on the corresponding stratum of the bordification, and in that deformation space it may be irreducible (sometimes this is referred as relative irreducibility).

The symplectic structure for renormalization of circle diffeomorphisms with breaks

2021 ◽  
pp. 2150016
Author(s):  
S. Ghazouani ◽  
K. Khanin
Keyword(s):  
Moduli Spaces ◽  
Symplectic Form ◽  
Symplectic Structure ◽  
Mapping Class ◽  
Character Variety ◽  
Parabolic Bundles ◽  
Dimension Formula ◽  
Renormalization Operator ◽  
Symplectic Dynamics ◽  
Circle Diffeomorphisms

The main goal of this paper is to reveal the symplectic structure related to renormalization of circle maps with breaks. We first show that iterated renormalizations of [Formula: see text] circle diffeomorphisms with [Formula: see text] breaks, [Formula: see text], with given size of breaks, converge to an invariant family of piecewise Möbius maps, of dimension [Formula: see text]. We prove that this invariant family identifies with a relative character variety [Formula: see text] where [Formula: see text] is a [Formula: see text]-holed torus, and that the renormalization operator identifies with a sub-action of the mapping class group [Formula: see text]. This action allows us to introduce the symplectic form which is preserved by renormalization. The invariant symplectic form is related to the symplectic form described by Guruprasad et al. [Group systems, groupoids, and moduli spaces of parabolic bundles, Duke Math. J. 89(2) (1997) 377–412], and goes back to the earlier work by Goldman [The symplectic nature of fundamental groups of surfaces, Adv. Math. 54(2) (1984) 200–225]. To the best of our knowledge the connection between renormalization in the nonlinear setting and symplectic dynamics had not been brought to light yet.

WHEN IS THE DEFORMATION SPACE $\mathscr{T}(\Gamma, H_{2n+1}, H)$ A SMOOTH MANIFOLD?

2011 ◽  
Vol 22 (11) ◽  
pp. 1661-1681 ◽  
Author(s):  
ALI BAKLOUTI ◽  
SAMI DHIEB ◽  
KHALED TOUNSI
Keyword(s):  
Smooth Manifold ◽  
Hausdorff Space ◽  
Necessary Condition ◽  
Deformation Space ◽  
Deformation Spaces ◽  
Natural Action ◽  
Manifold Structure ◽  
Klein Form ◽  
Lie Subgroup ◽  
Sufficient And Necessary Condition

Let G = H2n + 1 be the 2n + 1-dimensional Heisenberg group and H be a connected Lie subgroup of G. Given any discontinuous subgroup Γ ⊂ G for G/H, a precise union of open sets of the resulting deformation space [Formula: see text] of the natural action of Γ on G/H is derived since the paper of Kobayshi and Nasrin [Deformation of Properly discontinuous action of ℤk and ℝk+1, Internat. J. Math.17 (2006) 1175–1190]. We determine in this paper when exactly this space is endowed with a smooth manifold structure. Such a result is only known when the Clifford–Klein form Γ\G/H is compact and Γ is abelian. When Γ is not abelian or H meets the center of G, the parameter and deformation spaces are shown to be semi-algebraic and equipped with a smooth manifold structure. In the case where Γ is abelian and H does not meet the center of G, then [Formula: see text] splits into finitely many semi-algebraic smooth manifolds and fails to be a Hausdorff space whenever Γ is not maximal, but admits a manifold structure otherwise. In any case, it is shown that [Formula: see text] admits an open smooth manifold as its dense subset. Furthermore, a sufficient and necessary condition for the global stability of all these deformations to hold is established.

OPTIMIZED DESCRIPTION OF NUCLEAR SHAPES AND SYMMETRIES

2007 ◽  
Vol 16 (02) ◽  
pp. 541-551 ◽  
Author(s):  
ANDRZEJ GÓŹDŹ ◽  
JERZY DUDEK
Keyword(s):  
Group Theory ◽  
Large Scale ◽  
Computing Time ◽  
Mean Field ◽  
Constraint Equation ◽  
Deformation Space ◽  
Deformation Spaces ◽  
Hartree Fock ◽  
Deformation Parameters

We propose a group-theory-based method of analysis of the multipole type (αλμ) deformation spaces for the large scale calculations within nuclear mean-field applications. It allows to find ahead of time which sub-sets of the deformation space, although formally different, in fact represent the same information (the same nuclear shapes expressed by an alternative combination of the deformation parameters). The approach presented allows to save important amounts of the computing time in the large-scale mean-field calculations, both in constrained Hartree-Fock (where αλμ is replaced by the constraint equation [Formula: see text]), and in the Strutinsky type approaches.

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