Generalized q‐Dimensions of Measures on Non‐Autonomous Conformal Sets

The generalized ‐dimension of a measure provides information both on the multifractal structure of the measure and on the dimension of its support. In the self‐conformal setting, its value is obtained from the zero of a pressure function, but non‐autonomous attractors carry no invariant dynamics, so the ergodic‐theoretic tools are unavailable and the generalized ‐dimension may not exist. In this paper, we study measures supported on attractors of non‐autonomous conformal iterated function systems under a scale condition that permits the minimum contraction ratios to tend to zero. We define generalized upper and lower pressures through cut sets organized by the geometric scale, and we prove that their critical values bound the generalized ‐dimensions from above. Moreover, under the open set condition, these bounds are sharp and yield dimension formulae for every . For autonomous systems and , the generalized ‐dimension of a Gibbs measure is given by the unique zero of this pressure. Finally, we provide some examples to illustrate the autonomous and non‐autonomous cases, to clarify the roles of the separation and scale conditions, and to show that the scale condition cannot be dispensed with even under the strong separation condition.