The continued fraction is an alternative way to represent a real number as a limit of a sequence of rational numbers. Recall that for a real number α its continued fraction is an expression of the form
α = 

= a_{0}+1/(a_{1}+1/(a_{2}+1/(a_{3}+...))) = 
[a_{0}; a_{1}, a_{2}, a_{3}, ... ] 
1 
x 
[  1  ] 
x 
T: [0;a_{1},a_{2},a_{3}...] ↦ [0;a_{2},a_{3},a_{4}...]
It is therefore not invertible, but has infinitely many inverse branches T_{j} : x ↦1 
x + j 
T_{j} : [0;a_{1},a_{2},a_{3}...] ↦ [0;j,a_{1},a_{2},a_{3}...]
We may pick a pair of integers, say, 1 and 2 and consider an iterated function scheme of two inverse branches T_{1},T_{2}. Then for any α∈[0,1] = [0;a_{1},a_{2},a_{3}...] and for any sequence {j_{n}} of 1s and 2s the imagesT_{jn}...T_{j3}T_{j2}T_{j1} : α= [0;a_{1},a_{2},a_{3}...] ↦ [0;j_{n},...,j_{3},j_{2},j_{1},a_{1},a_{2},a_{3}...]
converge to a number whose partial denominators are all either 1 or 2. The set of all such numbers has been well studied and even has a dedicated notation.C_{2} = {x ∈ (0,1)  x = [0; a_{1}, a_{2}, a_{3}, ... ], where a_{j} = 1 or a_{j} = 2 for all indices j }.
It is a nonlinear Cantor set and it has a complex fractal structure. More generally, the sets which arise as limit sets of iterated function schemes of (a finite number of) the inverse branches T_{j} are called Gauss—Cantor sets. The Hausdorff dimension of C_{2} has now been computed to very high accuracy and we give 200 digits of it in Section 4.1.2 of our paper. For practical applications it is useful to know thatdim_{H} C_{2} = 0.53128... > 0.5.
Figure 1 on the left depicts the dimension of a twoparameter family of setsE_{n,k} = {x ∈ (0,1)  x = [0; a_{1}, a_{2}, a_{3}, ... ], where a_{j} ∈ {n, n+1, ... , n+k} for all indices j }
as a function of two variables d(n,k) = dim_{H} E_{n,k} for n,k = 1, ... , 50 . The set C_{2} corresponds to the choices n = 1 and k = 1, so C_{2} = E_{1,1}. The method we developed allows us to compute these 2500 values in just a few minutes with accuracy of 10^{−5}. We see that for any fixed k_{0} the function d(n,k_{0}) decreases rather rapidly as a function of n. For instance, in a special case k_{0} = 1 with only two digits n and n+1 in the expansion, we getd(2,1) = 0.3374... , d(3,1) = 0.2637... , d(50,1) = 0.0883....
On the other side of the parameter domain, when k_{0} = 50 we haved(1,50) = 0.9872..., d(2,50) = 0.8085..., d(50,50) = 0.4594....
Our algorithm enabled us to establish several estimates on the dimension of Gauss—Cantor sets, which are essential for the results on the densityone and modular forms of the Zaremba conjecture (pdf). Continued fractions also appear in Diophantine approximation in connection with the Markov and Lagrange spectra. The Markov spectrum first appeared in 1879–80 in work by A. A. Markov (father) on quadratic forms. Later, Perron gave an alternative definition, in terms of continued fractions. On a set of biinfinite sequences of natural numbers, we may define a realvalued functionλ : {a_{n}} ↦ a_{0} + [0;a_{1},a_{2},a_{3}...] + [0;a_{−1},a_{−2},a_{−3}...] ∈ R.
Then the Markov number of {a_{n}} is defined as a limit of the values of λ along the trajectory of the Bernoulli shift σ : {a_{n}} ↦ {a_{n+1}} , namelym({a_{n}}) := lim_{j→∞} λ(σ^{j}{a_{n}}).
The (classical) Markov spectrum M is then defined as the set of all Markov numbers for all possible sequences of integers. It is known that (∞,3) ∩ M is a discrete set and M ∩ (√21,+∞) is the entire ray. In the interval [3,√21] the Markov spectrum has rich fractal structure. Figure 2 on the left shows an approximation to the dimension of (∞,t) ∩ M as a function of t. A recent result by C. G. Moreira shows that d_{M}(t)=dim_{H}((∞,t) ∩ M) is a Cantor function, i.e. a continuous function which is locally constant on an open dense set. The approximation was calculated using the method we developed in our paper (joint with C. Matheus, C. G. Moreira and M. Pollicott), where we study other properties of the Markov spectrum and its closely related cousin called the Lagrange spectrum. The pink intervals show the gaps in the Markov spectrum, which are given, for instance, in the Cusick and Flahive’s textbook “Markov and Lagrange spectra” (published in 1991).PhD Thesis, University of Warwick,
under supervision of Dr. Oleg Kozlovski:
♜ Kinematic fast dynamo problem (pdf)
♜ Viva Talk Handout (pdf)
MRI Master Class project, University of Utrecht, under supervision of Prof. Yu. Kuznetsov:
♝ Accumulation of strong 1:2 resonances in generalised Hénon maps (pdf)
MSc Thesis, Independent University of Moscow, under supervision of Prof. G. Shabat:
♞ On nonarchimedean dynamical systems (pdf)