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Given a norm $\nu$ on $\mathbb{R}^2$, the set of $\nu$-Dirichlet improvablenumbers $\mathbf{DI}_\nu$ was defined and studied in the papers ofAndersen-Duke (Acta Arith. 2021) and Kleinbock-Rao (Internat. Math. Res.Notices 2022). When $\nu$ is the supremum norm, $\mathbf{DI}_\nu =\mathbf{BA}\cup \mathbb{Q}$, where $\mathbf{BA}$ is the set of badlyapproximable numbers. Each of the sets $\mathbf{DI}_\nu$, like $\mathbf{BA}$,is of measure zero and satisfies the winning property of Schmidt. Hence forevery norm $\nu$, $\mathbf{BA} \cap \mathbf{DI}_\nu$ is winning and thus hasfull Hausdorff dimension. In the present article we prove the followingdichotomy phenomenon: either $\mathbf{BA} \subset \mathbf{DI}_\nu$ or else$\mathbf{BA} \smallsetminus \mathbf{DI}_\nu$ has full Hausdorff dimension. Wegive several examples for each of the two cases. The dichotomy is based onwhether the critical locus of $\nu$ intersects a precompact $g_t$-orbit, where$\{g_t\}$ is the one-parameter diagonal subgroup of$\operatorname{SL}_2(\mathbb{R})$ acting on the space $X$ of unimodularlattices in $\mathbb{R}^2$. Thus the aforementioned dichotomy follows from thefollowing dynamical statement: for a lattice $\Lambda\in X$, either$g_\mathbb{R} \Lambda$ is unbounded (and then any precompact$g_{\mathbb{R}_{>0}}$-orbit must eventually avoid a neighborhood of $\Lambda$),or not, in which case the set of lattices in $X$ whose$g_{\mathbb{R}_{>0}}$-trajectories are precompact and contain $\Lambda$ intheir closure has full Hausdorff dimension.

A dichotomy phenomenon for Bad minus normed Dirichlet