Chaotic Dynamics of Conformable Semigroups via Classical Theory

Conformable derivatives involve a fractional parameter while preserving locality: on smooth functions they reduce to a classical derivative multiplied by an explicit weight. Exploiting this structural feature, we show that conformable time evolution does not give rise to a genuinely new semigroup theory. Rather, it can be fully interpreted as a classical $C_0$--semigroup observed through a nonlinear change of time. For $\delta\in(0,1]$, we introduce the conformable clock $$\Psi(t)=\frac{t^\delta}{\delta},$$ and prove that every $C_0$--$\delta$--semigroup $\mathcal S_\delta$ admits the representation $$\mathcal S_\delta(t)=\mathcal T(\Psi(t)),$$ where $\mathcal T$ is a uniquely determined classical $C_0$--semigroup on the same state space. This correspondence is exact at the infinitesimal level: the $\delta$--generator of $\mathcal S_\delta$ coincides with the generator of $\mathcal T$ on a common domain, and conformable mild solutions are in one-to-one correspondence with classical mild solutions under the reparametrization $s=\Psi(t)$. In particular, orbit sets are unchanged by the conformable clock, so orbit-based linear dynamical properties are invariant; $\delta$--hypercyclicity and $\delta$--chaos coincide with their classical counterparts. As an application, we derive a conformable version of the Desch--Schappacher--Webb chaos criterion by transporting the classical result. The analysis is carried out in conformable Lebesgue spaces $L^{p,\delta}$, which are shown to be isometrically equivalent to standard $L^p$ spaces, allowing a direct transfer of estimates and spectral arguments. Altogether, the results clarify which dynamical features of conformable models are intrinsic and which arise solely from a nonlinear change of time.