I have a new cryptography algorithm, what should I do? I created a new, unknown symmetric encryption algorithm. It looks like it's very fast and can be useful. I'm kind of independent researcher, hobbyist. Since years I'm interesting in Kakutani's problem and it's generalizations. And that's why I was abble to create it. I'm not a matematician with academic degree, what's more I even can not programming (I know only total basics of C), but I am very involved on the subject of the Kakutani's problem and its generalizations, I have known and read all major scientific publications since years ago on this topic. Here is an example of hash function which can be done with Kakutani's sequences and it's generalizations:
https://arxiv.org/pdf/1801.05079.pdf
As you can see it is possible to made a brilliant cryptographic/hash algoritm based on these functions, if you know how to use them. In recent years, more and more scientists are beginning to see the cryptographic potential in this field of mathematics (Kakutani's problem). So it's not just some easy topic, another stupid algoritm, it is based on one of the biggest unsolved math problem. Someone who will create algoritm based on pseudorandomness of Kakutani's sequences (and generalizations) can probably create very strong cryptographic tool.
Algoritm like this (I mean this one which I created) requires testing in couple areas (confusion and diffusion, differential cryptanalysis and so on), but I do not know cryptography so well to do this on my own (even don't know how to start). Simultaneously I have strong believe that my algoritm is something valuable and interesting.
What should I do? Do you think that such algorithm can be commercialized or it is possible to somehow made a profit on it (I also thought about a patent)? Like I said, it looks like algoritm runs faster than AES but I'm not abble to check everything on my own, for example confusion and diffusion of this algoritm is still a mystery to me (but I hope that algoritm will pass all tests). Maybe some of you guys work in cryptography and can give me some advice? Or maybe should I publish it somewhere on the internet and forget about profits or even authorship (because what can I do as nobody in the cryptographic industry)?
Do you think that such algorithm can be commercialized[...]?


This is very unlikely. AES is fast enough for most applications and where it isn't one usually uses (the cheap) hardware acceleration to fix that. Other common scenarios have the usually fast enough for them ChaCha stream cipher. This only leaves true edge cases on ultra-low power devices but this has active research going on with free schemes and some efforts from NIST, so a scheme with no proper analysis will not impress / interest most people.


  [Is it] possible to somehow [make] a profit on it (I also thought about a patent)?


Patenting a cryptographic algorithm has a great history of seeing it being unused for at least the lifetime of the patent and potentially afterwards as well due to standardization and implementations having already chosen other schemes which are good enough. Examples of this are the OCB mode of operation which is faster than most other known authenticated encryption algorithms but has seen little use due to patents and Schnorr signatures which are only now seeing standardization - after the patents expired which in fact prompted (EC)DSA to be created to avoid the legal issues.


  it looks like algoritm runs faster than AES


Without proper security analysis speed claims will likely not be interesting. After all 4-round AES also runs faster than AES but is somewhat easily broken.


  Maybe some of you guys work in cryptography and can give me some advice? Or maybe should I publish it somewhere on the internet and forget about profits or even authorship (because what can I do as nobody in the cryptographic industry)?


While some active users here do work in cryptography, note that Crypto.SE does not accept questions to analyze non-trivial primitives as this cipher most likely would be. Publishing it and hoping someone will have a look at it because they find it interesting may be your best bet (sci.crypt and reddit's /r/crypto may be starting points for advertisement) though paying a professional cryptographer is the only "sure" way to get proper attention at the scheme.

Having a reputation in the academic community for cryptanalysis would indeed also help in getting more interest - especially as good schemes by people without reputation are indistinguishable from bad schemes by people without reputation with the latter being far more likely. Additionally most people don't react too well to their pet project being broken which means such analysis takes a lot of time and yields no satisfying result for anybody involved most of the time which is why most people don't do it (for free).



What could make this scheme potentially interesting is if it had a security reduction from (S)PRP security to some known / proven pseudorandomness result of the Collatz problems - but even then probably mostly for theoreticians.