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 $$\mathrm{\Psi }(t)=\frac{t^{\delta }}{\delta },$$ and prove that every $C_0$–$\delta$–semigroup ${\mathcal{S}}_{\delta }$ admits the representation $${\mathcal{S}}_{\delta }(t)=\mathcal{T}(\mathrm{\Psi }(t)),$$ 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 $s=\mathrm{\Psi }(t)$. 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.