To begin with let's look at what is known as the coastline paradox. Briefly it goes like this: If you measure the circumference of Britain with a 1km long stick, and then do the same with a 100m long stick, you will find that with the shorter stick you obtain a larger measurement. This will happen again if you use a 1m stick, the shorter the stick the longer the coastline. The simple reason of course is that the smaller stick can capture more detail.
Next imagine rising above the surface of Britain in a hot air balloon. At the lowest altitude you will see most detail. As you rise your image of the island will have less and less definition. Another thing you notice is that similar physical patterns appear, over and over, at different size scales. This phenomenon can be found all around us, even out into the universe at large and the very fabric of physics.
Now, keeping all that in mind, imagine progressing, from the big bang, through time, and looking at the overall complexity of the universe. {wild speculation follows} As time elapses we find increasing complexity evolving from simpler forms. More variety to its structure, more detail to its content (enter Life). Greater definition to the communications and even thoughts of beings like us.
At this point we should take stock of some core concepts. Firstly "entropy", that which we immediately call to mind when we think of the arrow time. These days Entropy is understood more in terms of energy dispersal, and equilibrium state than the older, order-disorder conception. The minimal composition whereof being that: As time progress (in the universe), less and less energy is available for doing work. This is of course the genesis of the heat death hypothesis for the Universe. What we need to be cognizant of here is: 1) global entropy always increase in a closed system, and we have no reason the classify the Universe as such. 2) Heat death is only one of several theories for the 'end' of the Universe.
Next let's think about 'complexity', it is in fact a distinct concept from order/disorder/chaos, the notable feature of which is that it increases over time. The apparent increase of overall, combined complexity, together with the observed proliferation of fractal structures, would suggest that the Universe has an overall fractal structure. But how could we leverage that?
There are certain metrics that can be employed in respect of fractals: for instance Kolmogorov complexity in regard of information complexity, and Hausdorff dimension relating to the iterative depth. The most promising one being fractal dimension which "...is an index for characterizing fractal patterns or sets by quantifying their complexity."
With the universe continually growing in complexity, the fractal dimension gives us a completely different means to measure Time. The usual way is by the interval of regular events. The new way evaluates the purely physical relation called the Fractal dimension, which gives us a time base fixed to the beginning of the Universe. An Absolute Time frame.
Question: Has this or anything similar been the subject of a serious treatise?
Bonus question: What would it mean if overall complexity starts diminishing?
Edit: Let me reiterate, the question is not about whether the universe has a fundamental fractal structure. The question is: What consequences would follow if the universe is fractal, specifically would it entail a universal time scale, starting from the origin of to universe, and measurable with fractal dimension. And of course, who has said what for or against it.
Why this question? An absolute time frame would have unimaginably far reaching impact on Science. (and with this my penchant for understatement has reach a new... extremity)
However it seems the notion that the universe could be fractal and/or that fractal dimension could be used as a measure of universal complexity, seems to big an obstacle for serious consideration of the question. So in this section I will try to make the premises a bit less ridiculous.
First a note on "complexity", it is different things to different scientists, as such it doesn't have a canonical definition in Science. But here we are talking about the entire universe, every conception of complexity must be taken into account. Here is an example from biology that links fractal dimension with complexity:
Fractal Dimension as a Quantitative Measure of Complexity in Plant Development John D. Corbit and David J. Garbary
Abstract
The shapes of 51 fronds from three species of brown algae (Fucus vesiculosus, Fucus serratus and Ascophyllum nodosum) were evaluated by computing the fractal dimensions (D) of their outlines. There was no difference in fractal dimension among mature fronds of the three species, and D was highly correlated with both developmental stage and structural complexity. With increasing age the plants grew not only larger but also more complex in form. Fractal dimension increased systematically with increasing complexity of shape from about 1 to 1.6. Fractal dimension thus provides a useful quantitative measure for the elaboration of shape complexity during plant development.
Note that complexity, and fractal dimension, increased with age Now recall "growth" is iterative, and the universe is growing.
https://www.thenatureofcities.com/2017/06/25/effect-iteration-urban-form-part/
So called 'fractals' are everywhere, to the extent that the word has been popularized. It now means almost any self similar hierarchical pattern. But they are not only geometric.
Fractal geometry, Turing machines, and divide-and-conquer recurrences [pdf] S. Dube, Informatique théorique et Applications/Theoietical Informaties and Applications
Quoted from: https://cstheory.stackexchange.com/q/16965
These results show that for every Turing machine there exists a fractal set which can be viewed, in a certain sense, as geometrically encoding the complement of the language accepted by the machine. One can build a fractal-based geometrical model of computation which is computationally universal. Secondly we survey the results which show how fractal geometry can be fruitfully used to solve divide-and-conquer recurrences. A recursive algorithm possesses temporal self-similarity and there is a natural connection with spatial self-similar objects (fractal images). This approach yields a new and gênerai way of solving such divide-and-conquer récurrences.
Fractals are in our bones and in our minds. They were expressed in the universe before Live began, are found at every scale and in every dimension...
It is no stretch to say that if a subsystem contains iterative structures its superior system should be the source. That is because in the self similar aspect of a fractal precludes subsystems from having new information. I'm not fan of Occam's razor but:
What is more plausible, a fractal universe or a universe of fractals?
Some further reading:
"We can now confidently state that nature seems fractal, but is that truly so?" - https://cosmosmagazine.com/physics/is-nature-really-chaotic-and-fractal-or-did-we-just-imagine-it
"By the volume measure, space is 3-D, but by the behavior of random motion, it is 1-D, or even a fractional dimension." - http://nautil.us/issue/29/scaling/the-case-for-fewer-dimensions