The time-scales theory combines differential equations theory with difference equations theory, and fractional calculus can provide more realistic models in practical problems. The fractional time-scales calculus has attracted much attention because it can unify continuous fractional systems and discrete fractional systems. Combining the time-scales calculus and the fractional calculus, we focus on the fractional time-scales Noether theorem with Caputo Δ-derivatives, which provides a new perspective for studying the dynamic behaviors of complex systems. This paper begins with a review of the definitions of fractional time-scales integrals and derivatives. Then, according to the proposed Caputo Δ-type fractional time-scales Hamilton principle, the fractional time-scales Lagrange equation is derived. Under certain conditions, the fractional time-scales Lagrange equation can be reduced to the time-scales Lagrange equation, the Caputo-type fractional Lagrange equation and the classical Lagrange equation. Furthermore, in the two cases of special infinitesimal transformations and general infinitesimal transformations, the definitions and criteria of Caputo Δ-type fractional time-scales Noether symmetries are given. In addition, the fractional time-scales Noether theorem under special infinitesimal transformations (Theorem 1) and the fractional time-scales Noether theorem under general infinitesimal transformations (Theorem 2) are proposed and proved. When
\alpha=1
, Theorem 1 can be reduced to the classical time-scales Noether theorem under special infinitesimal transformations, and Theorem 2 becomes the time-scales Noether theorem obtained by the generalized Jost method. Not only that, but Theorem 2 can be reduced to the Caputo-type fractional Noether theorem if
T=\mathbbR
. At the end of this paper, the fractional time-scales Kepler problem in the plane and the fractional time-scales single freedom linear vibration system are taken as examples to verify the correctness of theorems.
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