Textbook recommendations, please

I'm currently investigating an idea I've had, and in that context I need to familiarize myself with both local and non-local Fα-calculus ("fractal calculus") on fractal sets. I don't wish to go into the details of the project just yet, as right now it only exists in form of a very rough outline, and I need to to investigate first if it is in fact worthwhile pursuing at all (chances are it might not be). Suffice to say I need a mathematical toolset that generalizes ordinary multivariate calculus / differential geometry on smooth differentiable manifolds to fractal sets with non-integer Hausdorff dimensionality. So I'm looking for a text that introduces Fα calculus, including fractal derivatives and integrals, both of integer and fractional order (think Riemann-Liouville with fractal measure); a generalization of the usual differential operators (div, grad, curl,...) to fractal sets; differential equations on fractal sets; and ideally Dirichlet forms.

I've got access to "Fractal Calculus and Its Applications" by Golmankhaneh, but I find it to be too technical for me as an interested amateur. I'm hoping perhaps someone here can recommend a text on the subject that is more accessible and builds intuition, rather than just listing definitions and lemmas? I've tried searching the Interwebz of course, but there appears to be surprisingly little literature on this particular subject - or perhaps I just didn't search for it right.

Thanks in advance!

Edited Wednesday at 12:07 PM5 days by Markus Hanke

Hi Markus, I'm just going out so very quickly this might be a start for you

Chaos and Non Linear Dynamics

page 448 ff

Scaling and f(alpha) indices

Robert C Hilborn

Oxford University Press.

Sternberg has some stuff in his Dynamical systems book but I think you mqay have to find pieces of information in lots of different palces.

(Non linear) Dynamics is a key phrase.

I have some more to check so will post again later.

Let us know how you get on.

I cannot make any recommendations as the field, and literature, are mostly beyond my level, but you've made me curious.

Is this, in any way, related to the non-linearity of the EFEs, and the fact that they are extremely difficult to solve, usually employing linear approximations, which restrict their validity to a limited range of input values, and omit some other interesting results ?

7 hours ago, Markus Hanke said:

I've got access to "Fractal Calculus and Its Applications" by Golmankhaneh,

This book from the same publisher and about 40% cheaper is more elementary and may suit your need more.

https://www.worldscientific.com/worldscibooks/10.1142/12345#t=aboutBook

Its content coverage is roughly similar to the Oxford book I quoted or this one form Cambridge University press

Chaos in Dynamical Systems

Edward Ott

In Chapter 9 he calls f(alpha) the singularity spectrum and discusses the application to Thermodynamics.

It should be noted that the notation in this field is very new and very fluid.

What's a "multifractal"?

2 hours ago, KJW said:

What's a "multifractal"?

Ott Ch10 is entitled Many Dimensions and Multifractals.

In Relativity treatments start with one spatial dimension and then move on to 2, 3 and 4.

Fractal geometry is no different.

Remembering that the much of the activity in fractal analysis take place in phase space of some sort, not ordinary x,y,z geometrical space we define a multifractal as

An object with a multiplicity of fractal dimensions.

In ordinary continuous analysis we start with at least to variables to define a calculus eg x and y to get dy/dx etc.

We then define multivariable calculus as the calculus of 3 or more variables and the derivatives are partial and one needs 3 axes or dimensions to place or embed the system in.

A similar scheme takes place for fractal dimensions, except the 'derivatives' are all these fractal limits based on some 'measure' or another.

One such measure is called the box (or box counting) measure which counts the number of 'boxes' of side epsilon needed 'cover' the fractal set.

This echos the epsilon delta argument in continuous analysis calculus.

And a limit count is taken as the epsilon becomes smaller and smaller (as in ordinary calculus if such a limit exists)

Ott provides a very good motivation for multifractals in chapter 11 where he examines the Physics of Diffusion Limited Aggregation, Electrodeposition, Viscous fingering and how and why simple 2D linear partial diff equations (Laplace) lead to fractal behaviour at moving boundaries, thus developing the patterns we see in Nature.

8 hours ago, studiot said:

This book from the same publisher and about 40% cheaper is more elementary and may suit your need more.

Great, thanks :)

13 hours ago, MigL said:

Is this, in any way, related to the non-linearity of the EFEs, and the fact that they are extremely difficult to solve, usually employing linear approximations, which restrict their validity to a limited range of input values, and omit some other interesting results ?

Let us, for the time being, just say that I am curious as to what happens when you relax the notion of smoothness that underlies pretty much all our physical models. I’m also curious what would happen if dimensionality of space/time were allowed to vary with scale, even just minutely, and take on non-integer values in some regimes. I’ve also recently discovered the concept of the fractional (distinct from “fractal”) derivative, which naturally introduces a notion of non-locality into analysis, so I am curious as to that, too.

I want to first learn what the literature says about these things, and, once I’m a little familiar with the tools of the trade, experiment a little myself, insofar as I am able to. I do have something particular in mind, and yes, it’s to do with spacetime, but I don’t know yet if that is viable even in principle, so I won’t go into it just yet.

14 hours ago, studiot said:

Let us know how you get on.

I’m sure I will have a lot of questions along the way!

Edited Thursday at 03:56 AM5 days by Markus Hanke

1 hour ago, Markus Hanke said:

I’ve also recently discovered the concept of the fractional derivative

You may also be interested in the concept of the fractional Fourier transform. Actually, there is a whole bunch of discrete concepts that may be able to be made fractional.

Recently, I posted the following in another thread that may be of interest to you:

On 5/4/2026 at 12:05 AM, KJW said:

I found this YouTube video about "Hyperoperations" which may be of interest. I especially found the "Commutative hyperoperations" at around 16:32 to be interesting.

10 hours ago, studiot said:

Ott Ch10 is entitled Many Dimensions and Multifractals.

Sorry my mistake.

My posted table came from Ott ch9 multifractals.

Ch10 Many Dimensions and Multifractals is in Holborn.

(I picked up Holborn whilst Ott was left on the scanner)

5 hours ago, KJW said:

You may also be interested in the concept of the fractional Fourier transform. Actually, there is a whole bunch of discrete concepts that may be able to be made fractional.

Recently, I posted the following in another thread that may be of interest to you:

Thanks I've a good bit of material to go through now, much of it pretty non-trivial, so I'll have to take it a step at a time. This is all new territory to me, as until quite recently I wasn't aware that such things as fractal and fractional calculus even existed.

I guess Socrates and Einstein were right ...

"The more you learn, the more you realize how little you know"
-Socrates

"The more I learn, the more I realize how much I don't know"
-Einstein

10 hours ago, MigL said:

I guess Socrates and Einstein were right ...

"The more you learn, the more you realize how little you know"
-Socrates

"The more I learn, the more I realize how much I don't know"
-Einstein

Exactly.