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Алексей Канель-Белов:
Dear Colleagues,
Allow me to highlight several questions which appear after
Mehdi Golafshan's talk; in my opinion, this should be related to
our work in progress with Nikolai Germanovich Moshchevitin.
Please comment and add your questions.
The talk was about Sturm's words and discrete dynamics.
We consider words in some finite alphabet which satisfy certain
properties < Mehdi, please give exact definitions> concerning their
factors.
One of the key example was related to the golden ratio \alpha:
y= \alpha x
We study this line propagating from (0,0) for x>0,
and write down a sequence of 0 and 1 any time it crosses
the horizontal /resp. vertical line y\in Z (resp., x\in Z).
Question 1. What will happen if we consider
z= \alpha x, y= \beta x for different irrational \alpha, \beta?
How are ergodic sequences related to the problem we
study with N.G.Moshchevitin?
[In our work we consider approximation of several
independent irrational numbers by rational numbers
with bounded denomiators; we look at possible permutations;
here we look at sequences of words in some alphabet;
how are they related to permut
]
Question 2. What about approximating an "irrational"
2-plane by rational 2-planes? There are results by Schmidt;
Nikolai Germanovich can formulate them more exactly.
Can we define any "ergodic" properties corresponding to
2-planes in R^{4} (or any other dimension)?
Question 3. How is this theory related to the dynamics
if we take a geodesic line for a hyperbolic surface of genus
g glued from a (4g)-gon? We'll have 2g letters; how to describe
the set of words one can obtain this way? How are they
related to Sturm's word? How are they related to Gromov's work?
Question 4. a) What if we consider several "irrational"
geodesic lines? Can we get a presentation of the braid group
acting on the words?
b) Can we get representations of "non-closed braids"
by using some averaging;
maybe, we may tackle some asymptotic invariants like
Arnold's asymptotic linking number
c) Possibly, we can invent "non-closed analogues" of other
groups, not necessarily braid groups.
Question 5. How is the above theory related to the
distribution of primes?
Please upload all necessary papers: Belov-Mehdi-Mitrofanov,
Moshchevitin-Kahn etc.