arXiv subjects
Publications and source records attributed to Alok Kumar Yadav.
Any affine map on the (n+1)-dimensional Euclidean space gives rise to a natural map on the n-dimensional sphere whose dynamical aspects are not so well-studied in the literature. We explore the dynamical aspects of these maps by investigating about their distality and expansivity.
For a locally compact metrizable group $G$, we study the action of $\rm Aut(G)$ on $\rm Sub_G$, the set of closed subgroups of $G$ endowed with the Chabauty topology. Given an automorphism $T$ of $G$, we relate the distality of the $T$-action on $\rm Sub_G$ with that of the $T$-action on $G$ under a certain condition. If $G$ is a connected Lie group, we characterise the distality of the $T$-action on $\rm Sub_G$ in terms of compactness of the closed group generated by $T$ in $\rm Aut(G)$ under certain conditions on the center of $G$ or on $T$ as follows: $G$ has no compact central subgroup of positive dimension or $T$ is unipotent or $T$ is contained in the connected component of the identity in $\rm Aut(G)$. Moreover, we also show that a connected Lie group $G$ acts distally on $\rm Sub_G$ if and only if $G$ is either compact or it is isomorphic to a direct product of a compact group and a vector group. All the results on the Lie groups mentioned above hold for the action on $\rm Sub^a_G$, a subset of $\rm Sub_G$ consisting of closed abelian subgroups of $G$.
Consider the action of $GL(n,\mathbb{Q_p})$ on the $p$-adic unit sphere $\mathcal{S}_n$ arising from the linear action on $\mathbb{Q}_p^n\setminus\{0\}$. We show that for the action of a semigroup $\mathfrak{S}$ of $GL(n,\mathbb{Q}_p)$ on $\mathcal{S}_n$, the following are equivalent: (1) $\mathfrak{S}$ acts distally on $\mathcal{S}_n$. (2) the closure of the image of $\mathfrak{S}$ in $PGL(n,\mathbb{Q}_p)$ is a compact group. On $\mathcal{S}_n$, we consider the `affine' maps $\overline{T}_a$ corresponding to $T$ in $GL(n,\mathbb{Q}_p)$ and a nonzero $a$ in $\mathbb{Q}_p^n$ satisfying $\|T^{-1}(a)\|_p<1$. We show that there exists a compact open subgroup $V$, which depends on $T$, such that $\overline{T}_a$ is distal for every nonzero $a\in V$ if and only if $T$ acts distally on $\mathcal{S}_n$. The dynamics of `affine' maps on $p$-adic unit spheres is quite different from that on the real unit spheres.
Consider the action of $SL(n+1,\mathbb{R})$ on $\mathbb{S}^n$ arising as the quotient of the linear action on $\mathbb{R}^{n+1}\setminus\{0\}$. We show that for a semigroup $\mathfrak{S}$ of $SL(n+1,\mathbb{R})$, the following are equivalent: $(1)$ $\mathfrak{S}$ acts distally on the unit sphere $\mathbb{S}^n$. $(2)$ the closure of $\mathfrak{S}$ is a compact group. We also show that if $\mathfrak{S}$ is closed, the above conditions are equivalent to the condition that every cyclic subsemigroup of $\mathfrak{S}$ acts distally on $\mathbb{S}^n$. On the unit circle $\mathbb{S}^1$, we consider the `affine' actions corresponding to maps in $GL(2,\mathbb{R})$ and discuss the conditions for the existence of fixed points and periodic points, which in turn imply that these maps are not distal.