*Article* **"Holographic Implementations" in the Complex Fluid Dynamics through a Fractal Paradigm**

**Alexandra Saviuc 1,\*, Manuela Gîrt,u 2, Liliana Topliceanu 3, Tudor-Cristian Petrescu <sup>4</sup> and Maricel Agop 5,6**

	- <sup>4</sup> Department of Structural Mechanics, Gheorghe Asachi Technical University of Iasi, 700050 Iasi, Romania; tudor.petrescu@tuiasi.ro
	- <sup>5</sup> Department of Physics, Gheorghe Asachi Technical University of Ias,i, 700050 Iasi, Romania; m.agop@yahoo.com
	- <sup>6</sup> Romanian Scientists Academy, 050094 Bucharest, Romania
	- **\*** Correspondence: iuliana.saviuc@gmail.com

**Abstract:** Assimilating a complex fluid with a fractal object, non-differentiable behaviors in its dynamics are analyzed. Complex fluid dynamics in the form of hydrodynamic-type fractal regimes imply "holographic implementations" through velocity fields at non-differentiable scale resolution, via fractal solitons, fractal solitons–fractal kinks, and fractal minimal vortices. Complex fluid dynamics in the form of Schrödinger type fractal regimes imply "holographic implementations", through the formalism of Airy functions of fractal type. Then, the in-phase coherence of the dynamics of the complex fluid structural units induces various operational procedures in the description of such dynamics: special cubics with SL(2R)-type group invariance, special differential geometry of Riemann type associated to such cubics, special apolar transport of cubics, special harmonic mapping principle, etc. In such a manner, a possible scenario toward chaos (a period-doubling scenario), without concluding in chaos (nonmanifest chaos), can be mimed.

**Keywords:** differentiability; fractal hydrodynamic regimes; fractal Schrödinger regimes; fractal soliton; fractal kink; "holographic implementations"; cubics; apolar transport; harmonic mapping principle; period doubling scenario
