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We prove that the radii of convergence of the solutions of a$p$-adic differential equation $${\fancyscript{F}}$$ over an affinoid domain$X$of the Berkovich affine line are continuous functions on$X$that factorize through the retraction of $${X\to\Gamma}$$ of$X$onto a finite graph $${\Gamma\subseteq X}$$ . We also prove their super-harmonicity properties. This finiteness result means that the behavior of the radii as functions on$X$is controlled by a$finite$family of data.

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We showed that there is a complete analogue of a representation of the quantum plane ℬq where \q\ = 1, with the classical ax+b group. We showed that the Fourier transform of the representation of ℬ-q$ on ℋ = L2(ℝ) has a limit (in the dual corepresentation) toward the Mellin transform of the unitary representation of the ax+b group, and furthermore the intertwiners of the tensor products representation has a limit toward the intertwiners of the Mellin transform of the classical ax+b representation. We also wrote explicitly the multiplicative unitary defining the quantum ax+b semigroup and showed that it defines the corepresentation that is dual to the representation of ℬq above, and also correspond precisely to the classical family of unitary representation of the ax+b group.