Topological equivalence of flows on homogeneous spaces, and divergence of one-parameter subgroups of Lie groups

Abstract:

Let $\Gamma$ and $\Gamma ’$ be lattices, and $\phi$ and $\phi ’$ one-parameter subgroups of the connected Lie groups $G$ and $G’$. If one of the following conditions (a), (b), or (c) hold, Theorem A states that if the induced flows on the homogeneous spaces $G/\Gamma$ and $G’ /\Gamma ’$ are topologically equivalent, then they are topologically equivalent by an affine map. (a) $G$ and $G’$ are one-connected and nilpotent. (b) $G$ and $G’$ are one-connected and solvable, and for all $X$ in $L(G)$ and $X’$ in $L(G’ )$, $\operatorname {ad} (x)$ and $\operatorname {ad} (X’ )$ have only real eigenvalues, (c) $G$ and $G’$ are centerless and semisimple with no compact direct factor and no direct factor $H$ isomorphic to $\operatorname {PSL} (2, R)$ such that $\Gamma H$ is closed in $G$. Moreover, in condition (c), the induced flow of $\phi$ on $G/\Gamma$ is assumed to be ergodic. Theorem A depends on Theorem B, which concerns divergence properties of one-parameter subgroups. We say $\phi$ is isolated if and only if for any $\phi ’$ which recurrently approaches $\phi$ for positive and negative time, $\phi$ equals $\phi ’$ up to sense-preserving reparameterization. Theorem B(a) states that if $G$ is one-connected and nilpotent, or one-connected and solvable with exp: $L(G) \to G$ a diffeomrophism, then every $\phi$ of $G$ is isolated. Let $G$ be connected and semisimple and $\phi (t) = \exp (tX)$. Then Theorem B(b) states that $\phi$ is isolated, if $[X, Y] = 0$ and $\operatorname {ad} (Y)$ being semisimple imply that $\operatorname {ad} (Y)$ has some eigenvalue not pure imaginary and not zero. If $G$ has finite center, $\phi$ is isolated if there is no compact connected subgroup in the centralizer of $\phi$.