论文标题
$ p $ - adic球的组结构和球体上的动力学系统
Group structure of the $p$-adic ball and dynamical system of isometry on a sphere
论文作者
论文摘要
在本文中，研究了$ p $ - adic球和球体的组结构。研究了在不变球上定义的等轴测图的动力学系统。我们分别在球和球体上分别定义了二进制操作$ \ oplus $和$ \ odot $，并证明这套是相对于操作的紧凑拓扑的Abelian集团。然后，我们表明，具有正半径的任何两个球（球）作为组是同构。我们证明，在$ \ Mathbb z_p $中引入的HAAR度量也是任意球和球的HAAR度量。我们研究由球体上定义的等轴测图产生的动力系统，并表明任何不是固定点的初始点的轨迹都不是收敛的。我们研究了该$ p $ - 亚种动力学系统的急性性，相对于降低了球体的归一化HAAR测量。对于$ p \ geq 3 $，我们证明动态系统不是ergodic。但是对于$ p = 2 $，在某些情况下，动态系统可能是千古的。
In this paper the group structure of the $p$-adic ball and sphere are studied. The dynamical system of isometry defined on invariant sphere is investigated. We define the binary operations $\oplus$ and $\odot$ on a ball and sphere respectively, and prove that this sets are compact topological abelian group with respect to the operations. Then we show that any two balls (spheres) with positive radius are isomorphic as groups. We prove that the Haar measure introduced in $\mathbb Z_p$ is also a Haar measure on an arbitrary balls and spheres. We study the dynamical system generated by the isometry defined on a sphere, and show that the trajectory of any initial point that is not a fixed point isn't convergent. We study ergodicity of this $p$-adic dynamical system with respect to normalized Haar measure reduced on the sphere. For $p\geq 3$ we prove that the dynamical systems are not ergodic. But for $p=2$ under some conditions the dynamical system may be ergodic.
