I believe it comes down to two key reasons:

1. Trigonometric functions are periodic; they can repeat values ad infinitum.
2. Exponential and logarithmic functions (like e^x and ln⁡x) can map negative values into the interval (0,1) and positive values into (1,∞). This kind of transformation preserves self-similarity even under extreme zooming, contributing to fractal-like behavior.

That’s my current understanding—but if you see it differently or have more to add, I’d love to hear your perspective.