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**Dynamics of certain distal actions on spheres.**
*(English)*
Zbl 1420.37013

The notion of distality was introduced by Hilbert and was studied in different contexts.

In this paper, the authors consider the action of \(\mathrm{SL}(n+1,\mathbb{R})\) on \(\mathbb{S}^n\) arising as the quotient of the linear action on \(\mathbb{R}^{n+1}\backslash\{0\}\). They show that, for a semigroup \(\mathfrak{S}\) of \(\mathrm{SL}(n+1,\mathbb{R})\), the following properties are equivalent: (1) \(\mathfrak{S}\) acts distally on the unit sphere \(\mathbb{S}^n\), and (2) the closure of \(\mathfrak{S}\) is a compact group.

They also show that in the case that \(\mathfrak{S}\) is a closed semigroup, the above conditions are equivalent to the condition that every cyclic subsemigroup of \(\mathfrak{S}\) acts distally on \(\mathbb{S}^n\).

In the case of the unit circle \(\mathbb{S}^1\), the authors consider actions corresponding to maps in \(\mathrm{GL}(2,\mathbb{R})\). They study the conditions for the existence of fixed points and periodic points, which imply that these maps are not distal.

In this paper, the authors consider the action of \(\mathrm{SL}(n+1,\mathbb{R})\) on \(\mathbb{S}^n\) arising as the quotient of the linear action on \(\mathbb{R}^{n+1}\backslash\{0\}\). They show that, for a semigroup \(\mathfrak{S}\) of \(\mathrm{SL}(n+1,\mathbb{R})\), the following properties are equivalent: (1) \(\mathfrak{S}\) acts distally on the unit sphere \(\mathbb{S}^n\), and (2) the closure of \(\mathfrak{S}\) is a compact group.

They also show that in the case that \(\mathfrak{S}\) is a closed semigroup, the above conditions are equivalent to the condition that every cyclic subsemigroup of \(\mathfrak{S}\) acts distally on \(\mathbb{S}^n\).

In the case of the unit circle \(\mathbb{S}^1\), the authors consider actions corresponding to maps in \(\mathrm{GL}(2,\mathbb{R})\). They study the conditions for the existence of fixed points and periodic points, which imply that these maps are not distal.

Reviewer: Marta Macho Stadler (Leioa)

### MSC:

37C85 | Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\) |

37B05 | Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) |

37C25 | Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics |

54H20 | Topological dynamics (MSC2010) |

43A60 | Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions |