A φ-Contractivity and Associated Fractal Dimensions

In this paper, we extend the concept of dimension of sets to some general frameworks relative to a gauge function φ, where two simultaneous dimensions are introduced. Unlike the classical cases where one dimension function is introduced based on the diameter power relative to the associated measure power, and where the gauge is a set-valued function or a measure in the majority of cases, we no longer assume this hypothesis. The introduced variant generalizes many existing cases, such as Haudorff, packing, Carathéodory, and Billingsley original variants. Many characteristics of the dimensions are investigated, such as bijectivity, convexity, monotony, asymptotic behavior, and fixed points.