Thank you Darkdoji for your reply.
Although I like the insights from fractals and chaos as a mathematician, I haven't yet seen anything concrete for practical trading.
I respect Williams, and his "fractals" have some usefulness, but they have nothing to do fractals in Mathematics.
IFS are good for generating fractal objects (e g ferns, etc) but I doubt if the market is deterministic. By the way, the paper I mentioned before does not mention IFS.
Prof Didier Sornette is an expert on multifractals, I believe he is working on using multifractals to detect forthcoming market bubbles. This could be very useful.
Peters with his Fractal Market hypothesis (FMH) sees the market more like a stochastic process (not deterministic), but where the probability distributions are not normal. In FMH outliers are "normal".
I think Mandelbrot agrees that the market is random if you read the article on "the Seven states of randomness" ( https://en.wikipedia.org/wiki/Seven_sta ... randomness
These are my views.
Now to respond more directly to the points you make here.
I take your first point to imply that you have not read my presentation "the chaology of markets" before arriving at that conclusion. If on the other hand you have (and I believe we did a clear job of stating exactly how the market works by chaos and by fractals) I shall be grateful (for a mathematician as you are) to point out where in that presentation I have adduced material that confirms your current position? Is there any point that was not covered to make the presentation not the joined-up thinking it was intended to be? I shall be most grateful and enlightened by your views.I have no regard for what Bill Williams thinks and consider him strictly a technical analyst (regardless of whatever so far unexpressed knowledge of chaos and fractals he may have). It is this point you made that indicates to me that you may not have read my output yet. For a mathematician to associate even remotely my ideas with Bill Williams (after reading what I have had to say in the chaology) is simply not possible. As such I can only assume you are speaking without reference to my stated position. Please read the material if you do not mind.Again here you miss the point. What is a bubble? How much of that is your concern from day - day as a trader? I do not doubt the expertise you claim for Prof Didier Sornette and I wish him all the luck in the world. But frankly, the topics are different and whatever the good prof. comes up with will have to be evaluated with the basic setup of markets which we have declared in the chaology indisputably (unless of course you can guide us as to where our math is wrong or misleading). Do not forget this is where I think Mandelbrot's profound knowledge as he has shared, is misplaced; trying to address quants and risk managers in the control of risk when in fact it is only by the trader (and his understanding of how the market works) that risk can be reduced. This is why we have provided a general multifractal trading model to any type of trader - algorithmic or otherwise to use at this time. Ultimately, fractals as it applies to the market is done and gone. Mandelbrot provided all we need to know and to the level that low level intellectuals like you and I can understand and deploy right now (I only hold 3 certificates of which the highest is a mere masters from some Scottish university). Bubbles do not matter much to me in the Gold (and sometimes currency markets) I care to trade. But I understand the sentiment - approval from academia is what we must live by and I do not argue with that - except to say that when that time comes (and if we are still alive) it will not dispute a dot of what we have described in the chaology.Again we can ignore (for the present purposes) Peters and his hypothesis. He like most academics and near academics have needs not shared by traders. As a mathematician I am sure you appreciate the extent to which mathematics is not arithmetic and as such you would know that much depends on the assumptions made whatever the math applied. For instance, in algorithmic trading, which I am very interested in as a prop trader, Glosten and Milgrom as extended by Samnay Das posit alpha to indicate the proportion of Informed Traders in a flow. These are traders who according to them know all future points all of the time and exactly (100% of the time). The rest of course are assumed to be uninformed traders who are as lost as the informed traders are gods in this setup. Now the evidence is clear that there is in fact no basis in reality to make such assumptions except for the simple convenience of making the ensuing calculation "portable" within the mathematical formalisms adopted for the calculations. GS, CiTi, Saxo or anyone else does not fit the bill of the know it all trader nor do the profits of such assumed characters (in real life) reflect that reality. But there are thousands of trades made based on exactly that thinking (baseless as it is) and this reflects (not surprisingly) exactly - the thinking of the pagan trader who assumes the market to be fixed. But what is worse, although the proportion of informed and uninformed traders in the market is assumed to be known, no details are given about how to calculate this property (so as to even glean the realism in that assumption). Strange is it not? Then inter alia, the model goes on to distribute probabilities according to the normal curve (and all that is implied by that for how price moves). So we have the sort of rigor I was taught to get a masters but so what? The model fails and does so badly (as in fact it trades with no more understanding than we human traders have about the market). My point is lets not mix it up. The trader needs to understand how price moves in real time to calculate a response. Mandelbrot already provided that and all I have done is to show how it would work in electronically traded markets - if we find that true then we can always measure what others are saying against such an understanding (Peters in his hypothesis introduces an assumption so similar to the one we just discussed that I worry about. Mainly because everyone (except Mandelbrot) just tries to avoid the market in terms of point displacement as the mechanism to understand which is what fractal geometry is setup to do. And that is how traders can read the markets, so it is fair to say Peters is not talking to traders and so we can ignore peters (I have) on that score as traders).IFS is mentioned in the paper you highlighted and very elaborately too (i.e. to detail). The whole point of those elaborate diagrams Mandelbrot was so eager to show us all is IFS. You being a mathematician can easily check this with any professor of your choice anywhere on earth. I take it that you have not had more than a passing interest in fractal geometry not to know that. But please correct me if I am wrong; in the chaology, we show you how and why the IFS is pertinent in any approach that seeks to understand how price moves. It should be easy then to dispute my findings simply by showing that what I describe cannot be. But if you imply "not mentioned by name" then consider the book "The (Mis) Behavior" of markets by Mandelbrot and another, where the same material is represented and you should then have no doubts. Finally, Mandelbrot CANNOT disclaim his own thesis can he? You say elsewhere that "where the probability distributions are not normal" and acquiesce or at least do not dispute that assertion, then how can Mandelbrot "think" as you assert? Fractals are distinguished by being deterministic variables that appear random in chaotic space and we make clear why that is in the chaology. Again being a mathematician it is more than simple for you to affirm this with a relevant math professor anywhere on earth. With regard to that point then - I can only say, I think you are very wrong and may have looked at the material you cite - only sparingly.So all in all - those are my responses and I am happy to clarify any areas not immediately clear (I am sorry my English is completely inadequate so there may be areas that would not be clear to a native speaker if you are). In any case, I hope you will respond so that we can continue this already very interesting discussion. Thanks so much for your time (why? Because you help to validate my ideas and prove its worth to all interested in breaking the gridlock we have had in trading for so long).