Symmetries and reversing symmetries of polynomial automorphisms of the plane

可接受的反向和扩展对称性用于双射替换

Admissible reversing and extended symmetries for bijective substitutions

本文研究了双射替换生成的移位的反向和扩展对称性，提出了判断条件和算法，并展示了如何构造具有特定对称群的无周期包络。

Recognition of symmetries in reversible maps

We deal with germs of diffeomorphisms that are reversible under an involution. We establish that this condition implies that, in general, both the family of reversing symmetries and the group of symmetries are not finite, in contrast with continuous-time dynamics, where typically there are finitely many reversing symmetries. From this we obtain two chains of fixed-points subspaces of involutory reversing symmetries that we use to obtain geometric information on the discrete dynamics generated by a given diffeomorphism. The results are illustrated by the generic case in arbitrary dimension, whe…

A brief guide to reversing and extended symmetries of dynamical systems

The reversing symmetry group is a well-studied extension of the symmetry group of a dynamical system, the latter being defined by the action of a single homeomorphism on a topological space. While it is traditionally considered in nonlinear dynamics, where the space is simple but the map is complicated, it has an interesting counterpart in symbolic dynamics, where the map is simple but the space is not. Moreover, there is an interesting extension to the case of higher-dimensional shifts, where a similar concept can be introduced via the centraliser and the normaliser of the acting group in the…

Normal forms of $ω$-Hamiltonian vector fields with symmetries

In this paper, we present algebraic tools to obtain normal forms of $ω$-Hamiltonian vector fields under a semisymplectic action of a Lie group, by taking into account the symmetries and reversing symmetries of the vector field. The normal forms resulting from the process preserve the Hamiltonian condition and the types of symmetries of the original vector field. Our techniques combine the classical method of normal forms of Hamiltonian vector fields with the invariant theory of groups.

遍历变换中心化子与本质非紧图对称性

Ergodic-transformation centralizers and essentially non-compact graphing symmetry

该研究证明无限标准概率空间上遍历变换的自同构群及其反向自同构群可实现为图的对称群，回答了Lovasz的问题并构造了具高度可变对称性的图族。

Reeb components with complex leaves and their symmetries I : The automorphism groups and Schröder's equation on the half line

We review the standard Hopf construction of Reeb components with leafwise complex structure and determine the group of leafwise holomorphic smooth automorphisms for tame Reeb components in the case of complex leaf dimension one. For this, we solve the Schröder type functional equation on the half line for expanding diffeomorphism. As a result, we see that the automorphism group of one with trivial linear holonomy on the boundary contains an infinite dimensional vector space, while in the case of non-trivial linear holonomy the group is of finite dimensional.

Topological symmetries of simply-connected four-manifolds and actions of automorphism groups of free groups

Let $M$ be a simply connected closed $4$-manifold. It is proved that any (possibly finite) compact Lie group acting effectively and homologically trivially on $M$ by homeomorphisms is an abelian group of rank at most two. As applications, let $\mathrm{Aut}(F_{n})$ be the automorphism group of the free group of rank $n.$ We prove that any group action of $\mathrm{Aut}% (F_{n})$ $(n\geq 4)$ on $M\neq S^{4}$ by homologically trivial homeomorphisms factors through $\mathbb{Z}/2.$ Moreover, any action of $% \mathrm{SL}_{n}(\mathbb{Q})$ $(n\geq 4)$ on $M\neq S^{4}$ by homeomorphisms is trivial.

具有反向定向对称性的 MHD 平衡的闭合轨道

Closed orbits of MHD equilibria with orientation-reversing symmetry

本文研究具有非反射型反向定向对称性的 MHD 平衡，利用拓扑方法证明在压力函数具有环形嵌套水平集时环面上所有轨道均周期，并提出刻画微分同胚作用的指标。

Automorphisms of generic gradient vector fields with prescribed finite symmetries

Let $M$ be a compact and connected smooth manifold endowed with a smooth action of a finite group $Γ$, and let $f$ be a $Γ$-invariant Morse function on $M$. We prove that the space of $Γ$-invariant Riemannian metrics on $M$ contains a residual subset ${\mathcal Met}_f$ with the following property. Let $g\in{\mathcal Met}_f$ and let $\nabla^gf$ be the gradient vector field of $f$ with respect to $g$. For any diffeomorphism $ϕ$ of $M$ preserving $\nabla^gf$ there exists some real number $t$ and some $γ\inΓ$ such that for every $x\in M$ we have $ϕ(x)=γ\,Φ_t^g(x)$, where $Φ_t^g$ is the time-$t$ fl…

Computation of extended symmetry groups for multidimensional subshifts with hierarchical structure

Automorphism groups are intrincate conjugacy invariants for subshifts, which can reveal important features of the dynamical structure of a shift action. One important case is the study of automorphism groups when the underlying subshift has a very rigid structure, e.g. substitutive subshifts or aperiodic constructions with large-scale self-similarity, such as the Robinson shift. In this work we study the automorphism group of bijective substitutive subshifts, and a potential generalization in the form of the group of extended symmetries, studied previously by Michael Baake, John Roberts and Re…

Periodic orbits in Hamiltonian systems with involutory symmetries

We study the existence of families of periodic solutions in a neighbourhood of a symmetric equilibrium point in two classes of Hamiltonian systems with involutory symmetries. In both classes, involutions reverse the sign of the Hamiltonian function. In the first class we study a Hamiltonian system with a reversing involution R acting symplectically. We first recover a result of Buzzi and Lamb showing that the equilibrium point is contained in a three dimensional conical subspace which consists of a two parameter family of periodic solutions with symmetry R and there may or may not exist two fa…

Compact Kähler manifolds with automorphism groups of maximal rank

For an automorphism group G on an n-dimensional (n > 2) normal projective variety or a compact Kähler manifold X so that G modulo its subgroup N(G) of null entropy elements is an abelian group of maximal rank n-1, we show that N(G) is virtually contained in Aut_0(X), the X is a quotient of a complex torus T and G is mostly descended from the symmetries on the torus T, provided that both X and the pair (X, G) are minimal.

数域中无幂整数及其移位空间的对称性

Symmetries of power-free integers in number fields and their shift spaces

本文研究了数域K整数环中保持k次无幂整数集合V_{K,k}的Z线性自同构群，证明其由域自同构与单位元乘法复合而成，并揭示了相关移位空间D_{K,k}的扩展对称群生成方式及系统间的拓扑非共轭性，最后将此概念推广至筛法。

Symmetries of the transfer operator for $Γ_0(N)$ and a character deformation of the Selberg zeta function for $Γ_0(4)$

The transfer operator for $Γ_0(N)$ and trivial character $χ_0$ possesses a finite group of symmetries generated by permutation matrices $P$ with $P^2=id$. Every such symmetry leads to a factorization of the Selberg zeta function in terms of Fredholm determinants of a reduced transfer operator. These symmetries are related to the group of automorphisms in $GL(2,\mathbb{Z})$ of the Maass wave forms of $Γ_0(N)$ . For the group $Γ_0(4)$ and Selberg's character $χ_α$ there exists just one non-trivial symmetry operator $P$. The eigenfunctions of the corresponding reduced transfer operator with eigen…