# Besicovitch cascades and bounded partial quotients

Andrey Kochergin 1
• 1 Lomonosov Moscow State University, , Russia
Andrey Kochergin

## Abstract

The article continues a series of works studying cylindrical transformations having discrete orbits (Besicovitch cascades). For any γ ∈ (0,1) and any ɛ > 0 we construct a Besicovitch cascade over some rotation with bounded partial quotients, and with a γ–Hölder function, such that the Hausdorff dimension of the set of points in the circle having discrete orbits is greater than 1 − γɛ.

Let 𝕋 = ℝ/ℤ be a circle of length 1, Tρ : 𝕋 → 𝕋 be an irrational circle rotation
$Tρx=x+ρ (mod1),$
and let f : 𝕋 → ℝ be a continuous function with zero mean. We consider a cylindrical transformation Tρ,f : 𝕋 × ℝ → 𝕋 × ℝ with a cocycle f :
$Tρ,f(x,y)=(Tρx,y+f(x)),$
which is a skew product over the circle rotation. (D.V. Anosov  also called it a cylindrical cascade.) The iterations of a point (x,y) are described by
$Tρ,fr(x,y)=(Tρrx,y+fr(x)), r∈ℤ,$
where fr (x) is a Birkhoff sum
$fr(x)={f(x)+f(Tρx)+…+f(Tρr−1x)for r>0,0for r=0,−f(Tρ−1x)−…−f(Tρr+1x)−f(Tρrx)for r<0.$

Cylindrical transformation was studed by H. Poincaré  as a model for flat transformations. In particular, he studied the ω-limit points of orbits. Using these mappings, he, as well as other authors, constructed examples of flows in spaces of higher dimension with various topological properties. Cylindrical transformation have various applications in ergodic theory: see, for example, , , .

It turned out that there are cascades having unbounded orbits, and there are those only with bounded ones. If f is a coboundary over Tρ in the class of continuous functions, i.e., there exists FC(𝕋) such that
$f(x)=F(Tρx)−F(x)$
for all x, then the orbit of each point (x0,y0) is contained within a closed invariant curve
${(x,y):y=y0+F(x)−F(x0), x∈T},$
and therefore it is bounded. In this case, the cylinder 𝕋 × ℝ splits into such curves, and Tρ,f is isomorphic to the rotation of the cylinder around the axis.

Later L.G. Shnirel’man  and A.S. Besicovitch  found examples of topologically transitive cylindrical cascades, i.e., cascades having dense orbits in the cylinder (such orbits are also called topologically transitive). At that time, they did not know whether all the orbits in this case could be topologically transitive.

In 1955 W.H. Gottschalk, G.A. Hedlund  showed that Tρ,fis topologically transitive if and only if f is not a coboundary over Tρand has zero mean.

The coboundaries and the cohomological (or homological in earlier terminology) equation (1) are of particular importance in the theory of dynamical systems. They are used to establish isomorphism of skew products, special flows, to study the spectral properties of metric automorphisms, etc. Cohomological equations can be considered in various classes of functions (see, for example, ). The Birkhoff sums for coboundaries have a convenient description
$fr(x)=F(Tρrx)−F(x).$
In 1951 A.S. Besicovitch  proved that a cylindrical cascade cannot be minimal, it has nontransitive orbits, and for any irrational circle rotation Tρ, there exists a continuous f such that Tρ,f is topologically transitive and has discrete orbits (which are closed invariant sets). The point (x,y) ∈ 𝕋 × ℝ has a discrete orbit if and only if lim
$lim|r|→+∞fr(x)=∞.$
The condition
$∫Tf(x) dx=0$
is essential. Otherwise, by ergodic theorem, fr(x) → ∞ as r → ± ∞, and so each orbit is discrete.
The presence of discrete orbits, as well as the property of a function to be a coboundary, substantially depends on the properties of ρ. Recall the basic notions. Every irrational ρ ∈ (0,1) may be represented as an infinite continued fraction
$ρ=1k1+1k2+⋯,$
or briefly,
$ρ=[k1,k2,…],$
and the natural numbers kn are called partial quotients. The fraction pn/qn = [k1,..., kn] is called the convergent of continued fraction. It is known , that the convergents pn/qn are determined recursively:
$pn+1=kn+1pn+pn−1, n≥1, p0=0, p1=1,qn+1=kn+1qn+qn−1, n≥1, q0=1, q1=k1.$
ρ is called Diophantine if there exist C > 0, θ > 0 such that $qn+1 for any n. Otherwise, it is called Liouville.

In 2010, K. Frączek and M. Lemańczyk  began to study the properties of a set of discrete orbits depending on the function f and the rotation number ρ. (Following them, a transitive cascade with discrete orbits is called the Besicovitch cascade, and the set B ⊂ 𝕋 × {0} of circle points having discrete orbits is called the Besicovitch set.)

The Besicovitch set B is invariant under Tρ, Tρ is uniquely ergodic with the only invariant Lebesgue measure, and, therefore, B has a null Lebesgue measure.

Obviously, if f has bounded variation, then Tf is not Besicovitch, because the sequence $Tρ,fqn(x,y)$ is bounded for denominators qn of ρ, as ‖ fqnC(𝕋) ⩽ Var(f).

K. Frączek and M. Lemańczyk showed that the Besicovitch cascade with continuous function f can be constructed for any transitive Tρ. For ρ satisfying some Diophantine condition, the γ-Hölder function f was obtained, so that Tρ,f is Besicovitch. In this construction, γ depends on the Diophantine parameter and γ < 1/2 in any case. They also showed that, under additional conditions, the Hausdorff dimension of the Besicovitch set can be at least 1/2.

Thus, it was established in  that both the admissible degree of continuity of the function and the Hausdorff dimension of the Besicovitch set depend on the properties of rotation.

We also note the result of E. Dymek , who showed that for any irrational ρ, one can construct a continuous cascade for which the Besicovitch set has a Hausdorff dimension 1.

A number of examples were constructed in , , demonstrating a closer relationship between the Hölder exponent of the function f and the obtained estimate of the Hausdorff dimension for the Besicovitch set. In , using angles with property $qn+1≍qn1+θ$ , θ > 0, it was shown that for any γ ∈ (0,1) and any ɛ > 0, there exist a γ-Hölder function f and a circle rotation Tρsuch that the cylindrical transformation Tρ,fis Besicovitch, and the Hausdorff dimension of the Besicovitch set in the circle is greater than 1 − γ− ɛ.

In , using angles with relatively slowly varying, but infinitely large partial quotients, it was managed to achieve inequality dimH(B) ⩾ 1 − γ and also to construct the Besicovitch cascade with a function that is γ-Hölder with any exponent γ ∈ (0,1).

Here we prove the following theorem.

Theorem 1

For any γ ∈ (0,1) and any ɛ > 0, there exists a γ-Hölder function f and a circle rotation Tρwith bounded quotients such that the cylindrical transformation Tρ,fis Besicovitch, and the Hausdorff dimension of the Besicovitch set in the circle is greater than 1 − γɛ.

Before proceeding to the proof of the theorem, we formulate two problems.

Problem 1

Is 1 − γ the upper bound for dimH(B)? It is unknown whether there exists a γ-Hölder Besicovitch cascade, for which the Hausdorff dimension of the Besicovitch set is greater than 1 − γ.

Problem 2

 Is it possible to construct a Besicovitch cascade with Hölder function over Liouville rotation of a circle?

The next three sections are devoted to the proof of the theorem.

## 2 The main construction

The construction is based on the design proposed in , but the operation with rotations having bounded partial quotients required a slight modification and more subtle estimates.

So, let ρ = [k1,...,kn,...] be an irrational number, and {pn/qn} be the sequence of convergents to ρ. Let δn = |ρpn/qn|. It is well known that 1
$1qn(qn+1+qn)<δn<1qnqn+1$
and
$ρ=pnqn+(−1)nδn.$
We will construct the γ-Hölder cocycle f as the sum of a series of Lipschitz functions fn corresponding to fractions pn/qn:
$f=∑n=1∞fn,$
where fn is 1/qn–periodical function defined by expression
$fn(x)={an2δn(x+δn)2,x∈[−δn,0],anδn2+anx,x∈[0,12tnqn−δn],an2tnqn−anδn4−anδn(x−12tnqn+δn2)2,x∈[12tnqn−δn,12tnqn],fn(1tnqn−δn−x),x∈[12tnqn,1tnqn],0,x∈[1tnqn,12qn−δn],−fn(x−12qn),x∈[12qn−δn,1qn−δn].$
In this expression, in addition to the rotation number parameters, the quantities tn and an are used. We assume, that for any number n
$tn>2.$
1/tn shows, what part of the period is occupied by the support of fn, and an > 0 is the Lipschitz constant for fn.

Note. Since the addition of the coboundary to the function $f=∑i=1∞fi$ does not change the Besikovich set, we can consider the terms and the convergents pn/qn, and also estimate the partial sums $∑i=n0nfir$ , starting from some number n0. For simplicity, in this case we shift the numbering, assuming n0 = 1.

Simultaneously, we define the subset of Besicovitch set D ⊂ 𝕋, each point of which has a discrete orbit:
$Gn+=∪k=0qn−1(kqn+[14tnqn,34tnqn]), Gn−=12qn+Gn+,Ds+=∩n=s∞Gn+, Ds−=∩n=s∞Gn−,D+=∪s=1∞Ds+, D−=∪s=1∞Ds−,D=D+∪D−.$
Lemma 2

If an ⩽ const(tnqn)1−γfor any n and some γ ∈ (0,1), then the series(4)converges, and the function f is γ–Hölder.

The detailed proof of this lemma is given in . The more general construction of functions with various continuity properties was also formulated and justified there. Earlier such construction of «almost Lipschitz function» was used by the author in .

The convergence of series (4) follows from the inequality $∥fn∥C⩽const(tnqn)γ$ and the fact that denominators qn grow not slower than exponentially.

$an=(tnqn)1−γ.$

The conditions of the next lemma are modified compared with those in the previous papers  and . These changes are adapted to dealing with bounded partial quotients.

Lemma 3
If the sequences {qn}, {tn} and {an} satisfy the conditions
$qn+1tnqn=m⩾6, m∈ℕ,$
$antn→+∞,$
$anqn+1tn2qn:an+1tn+1<16,$
then for any xD+
$limr→+∞fr(x)=+∞, limr→−∞fr(x)=−∞.$
For xD−, the situation is inverse.
Proof

1. At first, we will try to describe the mechanism of «pushing» the orbit of a point to infinity. The terms making up f, in turn, «are responsible» for the growth of Birkhoff sums on the set D+. Due to the good agreement with the circle rotation, the Birkhoff sum $fnr(x)$ of the n-th term for $x∈Gn+$ to a moment about r = qn/4 grows up to a level of the order an/tn, and at a moment of the order of $r=qn+1tn$ ceases to grow at a level no more than $Ln≈anqn+14tn2qn$ , and this value keeps constant up to the moment of at list qn+1/4, when the next term increases enough to «overtake» the current one. This fact explains the second and third assumptions of the lemma.

After the moment about r = qn+1/2, the Birkhoff sum $fnr(x)$ decreases and may become negative, but the growth of the sums fr on the set $Gn+1+∩Gn+$ is provided by the next summand $fn+1r$ . As each function fn is a coboundary, we can control the values of all previous terms, and the following terms are positive on D+.

2. We will show that fn is a coboundary. For this let us define a 1/qn–periodic function Fn : 𝕋 → ℝ such that
$fn(x)=Fn(Tρx)−Fn(x).$
We put
$F^n(x)={an2δnx2,x∈[0,12tnqn],Ln−F^n(1tnqn−x),x∈[12tnqn,1tnqn],Ln,x∈[1tnqn,12qn],Ln−F^n(x−12qn),x∈[12qn,1qn],$
where
$Ln=2F^n(1/(2tnqn))=an4δntn2qn2.$
It easy to verify that
$fn(x)=F^n(x+δn)−F^n(x).$
Then we put
$Fn(x)={F^n(x)for even n,Ln−F^n(x+δn)for odd n.$
For even n, when ρ = pn/qn + δn, we immediately have (11). For odd n, when ρ = pn/qnδn,
$Fn(x+ρ)−Fn(x)=−F^n(x+δn+ρ)+F^n(x+δn)=F^n(x+δn)−F^n(x+pn/qn)=fn(x).$
For the constant Ln, by (3) and (8), the estimates
$m4⋅antn
are valid.
3. Since for any integer r and any x ∈ 𝕋
$fnr(x)=Fn(x+rρ)−Fn(x), 0⩽Fn(x)⩽Ln,$
we have
$‖fnr‖C≤Ln.$
Moreover,
$|fnr(x)|⩽34Ln for any x∈D1± and any r∈ℤ,n∈ℕ.$
Indeed, if $x∈D1±$ , then $x∈Gn±$ , hence $14Ln⩽Fn(x)⩽34Ln$ , and so
$0−34Ln⩽fnr=Fn(Tρx)−Fn(x)⩽Ln−14Ln.$
It follows from (8) and (10) that
$antn:an+1tn+1<16m,$
therefore, by (13), we obtain
$L1+…+Ln
where, according to (5) and (8),
$K¯n=mini⩽nqi+1qi>12.$
Given the last three inequalities, we obtain
$L1+…+Ln
4. Now let be $x∈Gn+$ . Then
$fnr+1(x)⩾fnr(x)⩾0, fn−r(x)⩽0 for all r∈[0,qn+1/4],$
$|fn±r(x)|≥an18tn for all r∈[qn/4,qn+1/4].$
From (16) it follows that if $x∈D1+=∩n=1∞Gn+$ , then for all r ∈ [0,qn+1/4]
$fmr(x)⩾0 for m⩾n.$
To verify (16), note that by the periodicity of function fn and the set $Gn+$ ,
$fn(x+j/qn+ρ)=fn(x+(−1)nδn),$
therefore, we can assume that x ∈ [1/(4tnqn),3/(4tnqn)]. For even n, and given that (qn+1/4)δn < 1/(4qn), tn > 2 and $r⩽qn+14−1$ ,
$14tnqn⩽x+rδn⩽34tnqn+rδn<12qn−δn,$
so all steps $Tρrx$ within $r⩽qn+14−1$ occur in the region where fn is non-negative, which implies (16) and non-decreasing Birkhoff sums $fnr(x)$ with increasing r in the indicated range.

The cases of odd n and r < 0 are similar.

To get (17), we consider r = ⎡qn/4 ⎤. As above, we can assume that x ∈ [1/(4tnqn),3/(4tnqn)], and
$fnr(x)=∑k=0r−1fn(x+(−1)nkδn).$
By (3) and (8), points of the form {x + (−1)nn, 0 ⩽ kqn/4} fill evenly a segment of length shorter than 1/(4mtnqn) in the segment $[1−1/m4tnqn,3+1/m4tnqn]$ , and therefore
$fnr(x)⩾(1−12m)⋅an16tn.$
Roughing up the estimate, we obtain (17).
5. Now we can estimate Birkhoff sum fr(x) for $x∈D1+⊂Gn+$ and r ∈ [qn/4, qn+1/4]:
$fr(x)=∑k=1n−1fkr(x)+fnr(x)+∑k=n+1∞fkr(x)> {by (13),(18)} >−34(L1+…+Ln−1)+fnr(x)> {by (15),(17)} >−m4⋅an−1tn−1+118antn.$
According to (8) and (10), we have $m⋅an−1tn−1 , so
$fr(x)>antn(118−124)=172⋅antn→∞$
for n → ∞, and therefore, for r → +∞.

The cases r → − ∞ and $x∈D1−$ are similar. Cases $x∈Ds±$ , s > 1 require a shift in the numbering of terms.

Lemma 3 is proved.

## 3 Rotation of the circle

We consider an irrational number ρ with constant, starting from some position, partial quotients
$ρ=[Q,K,K,…],$
where Q is a rational number. In the standard representation with integer partial quotients, this means that partial quotients kn = K for all numbers n large enough. Thus, we define the sequence {qn} (and also {tn} and {an}). After that we fix the Hölder constant γ ∈ (0,1) and the constant
$m⩾6$
appearing in the construction of f. It will give us, by (8), $tn=qn+1mqn$ . To completely determine the cylindrical cascade Tρ,f, we define
$an=(tnqn)1−γ.$
So, we have three parameters K, γ and m, and we substitute them to the conditions of Lemma 3.
We have, starting from some number n,
$pn+1=Kpn+pn−1, qn+1=Kqn+qn−1.$
Since the sequence {qn} satisfies the difference equation, for n large enough we have
$qn=C1λn+C2(−λ)−n,$
where $λ=K2+K24+1≈K+1K$ is the root of characteristic equation λ2λ − 1 = 0. Also we have
$[K,K,K,…]=1λ, ρ=[Q,K,K,…]=1Q+1/λ.$
We say that anbn if an = bn (1 + O(1/λ−2n)) for n → ∞. In this notation
$qn≈Cλn, qn+1qn≈λ.$
Using equalities $qn+1tnqn=m$ (condition (8)) and an = (tnqn)1γ, we have
$tn=qn+1mqn≈λm,$
$an=(tnqn)1−γ≈(Cλm)1−γλn(1−γ),$
$antn≈C1−γ(mλ)γλn(1−γ),$
$anqn+1tn2qn:an+1tn+1=qn+1tnqn⋅(antn:an+1tn+1)≈mλ1−γ.$
Lemma 4
If the parameters γ, m, K are such that
$m⩾6, λ1−γ>6m,$
then the conditions of Lemma 3 are satisfied.
Proof

We used the condition (8) to determine tn. According to (24), the condition (10) of Lemma 3 holds for n large enough. By (21), the inequality tn > 6 holds for n large enough. Also, by (23), the condition (9), i. e., $antn→∞$ for n → ∞, is satisfied.

## 4 Hausdorff dimension of Besicovitch Set

In this section, we estimate the Hausdorff dimension of the Besicovitch set B, i. e., set of points on the circle 𝕋 × ℝ having discrete orbits. In section 2, the subset DB of Besicovitch set was constructed (see (6)):
$limr→±∞fr(x)=∞ for any x∈D.$
Thus, the lower bound for dimHD is also for dimHB. As for the upper bound for dimHD, the paper does not prove that it is such for dimHB, but such a hypothesis exists.
D is the union of a countable set of sets of Cantor type. It is sufficient to estimate $dimH(D1+)$ because
$dimH(D)=dimH(D1+).$
Lemma 5
The Hausdorff dimension of Besicovitch subset D satisfies
$ln(m2−1)lnλ⩽dimHD⩽ln(m2+1)lnλ.$
As a result,
$dimHB⩾ln(m2−1)lnλ.$

Recall, that $D1+=∩n=1∞Gn+$ (see (6)). Let us call the segments of $Gn+$ as n-th level segments.

We get the upper bound for dimHD by definition (see , section 2.1). Define the covers Wn of $D1+$ by segments inductively. $W1=G1+$ (we consider the cover as the union of segments). The cover Wn consists of those segments of $Gn+$ that intersect Wn1 in more than one point. The length of each of them is equal to 1/(2tnqn). By virtue of (8), each nth-level segment may intersect m/2 or m/2 + 1 segments of (n + 1)-th level. If mn is the number of nth–level segments included into Wn, than $mn⩽(m2+1)mn−1$ , or
$mn⩽m1(m2+1)n−1.$
Thus, we have the cover of $D1+$ by segments of length $1/(2tnqn)≈Cm⋅λn+1$ (by (20) and (5)). Then we have the estimate
$Hnd=mn⋅1/(2tnqn)d
If $λd>(m2+1)$ , then $limn→∞Hnd=0$ . Therefore, for Hausdorff dimension $d=dimHDn+$ , the inequality
$λd⩽(m2+1)$
is necessary, and we get the upper bound for Hausdorff dimension.
We obtain the lower bound for Hausdorff dimension by the method proposed in , Example 4.6. Now we denote by sn the minimal number of nth-level segments, contained in one (n − 1)th-level segment. Let ɛn be the minimal distance between n-th level segments. Then
$dimH(D1+)⩾liminfn→∞ln(s2…sn)−ln(sn+1εn+1).$
By (8), each n-th level segment contains entirely m/2 or (m/2 − 1) segments of the form [(i − 1) /qn+1, i/qn+1], therefore, each nth-level segment entirely contains at least (m/2 − 1) (n + 1)-th level segments, where from snm/2 − 1, we have $εn=1qn(1−12tn)$ . Substituting these inequalities and (20) to (29), we obtain
$dimH(D1+)⩾limn→∞(n−1)ln(m2−1)lnqn+1+lnsn+1−ln(1−12tn)=ln(m2−1)lnλ,$
and we get the lower bound in (26).
Lemma 6
For given γ ∈ (0,1) and any ɛ > 0, there exist an even integer m ⩾ 6, and the constant partial quotients K defining ρ, such that
$λ1−γ>6m and ln(m2−1)lnλ>1−γ−ε.$
Proof
Recall, that $λ=K2+K24+14≈K+1K$ . Since
$limλ→+∞λ1−γ/62(λ1−γ−ε+1)=+∞,$
the double inequality
$λ1−γ/6
which is equivalent to (30), is resolvable in the class of even m for any λ (and hence K) large enough.

This lemma completes the proof of Theorem 1.

In fact, for given γ ∈ (0,1) and ɛ > 0, we choose parameters m ⩾ 6 and K satisfying (30). Using these parameters and some rational Q, we define ρ = [Q,K,K,...] and thus a sequence of convergents {pn/qn}. Now, according to (8), the sequences {tn} and {an = (tnqn)1γ} are also defined.

Then, using the main construction, we define the function f and the set D.

According to Lemma 4, all the conditions of Lemma 3 are satisfied, therefore all points xD run away to infinity under iterations of the cylindrical cascade Tρ,f. So Tρ,f is a Besicovitch cascade, and D is a Besicovitch subset. By Lemma 2, f is γ-Hölder.

As m and K satisfy (25), from Lemma 5 we get the estimate
$dimHB>1−γ−ε.$

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E. Dymek, (2013), Transitive cylinder flows whose set of discrete points is of full Hausdorff dimension, arXiv: 1303.3099v1 [math.DS], 13 mar 2013.

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K. Frączek, M. Lemańczyk, (2010), On Hausdorff dimension of the set of closed orbits for a cylindrical transformation, Nonlinearity, 23, 2393–2422.

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W.H. Gottschalk, G.A. Hedlund, (1955), Topological Dynamics, Amer. Math. Soc. Colloq. Publ. 36, Amer. Math. Soc., Providence, RI, 1–148.

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A. Ya. Khinchin. (1964), Continued fractions, The University of Chicago Press, Chicago – London (transl. from Russian).

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A. Kochergin, (2002), A mixing special flow over a circle rotation with almost Lipschitz function, Sbornik: Mathematics, 193, 359–385.

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A. Kochergin, (2015), A Besicovitch Cylindrical Transformation with Hölder Function, Electronic Research Announcements in Mathematical Sciences, 22, 87 – 91. S 1935 – 9179 AIMS.

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A. Kochergin, (2018), New examples of Besicovitch transitive cylindrical cascades, Sb. Math., 209:9, 1257–1272.

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If the inline PDF is not rendering correctly, you can download the PDF file here.

• 

D. V. Anosov, (1973), On an additive functional homology equation connected with an ergodic rotation of the circle, Math. USSR-Izv., 7:6, 1257 – 1271

• Crossref
• Export Citation
• 

G. Atkinson, (1976), Recurrence of co-cycles and random walks, J. London Math. Soc., 13, 486–488.

• 

A.S. Besicovitch, (1937), A problem on topological transformation of the plane, Fund. Math., 28, 61–65.

• 

A.S. Besicovitch, (1951), A problem on topological transformations of the plane, Proc. Cambridge Philos. Soc., 47, 38–45.

• Crossref
• Export Citation
• 

E. Dymek, (2013), Transitive cylinder flows whose set of discrete points is of full Hausdorff dimension, arXiv: 1303.3099v1 [math.DS], 13 mar 2013.

• 

K. Falconer, (2003), Fractal geometry. Mathematical foundations and applications, John Wiley & Sons, Inc., Hoboken, NJ.

• 

K. Frączek, M. Lemańczyk, (2010), On Hausdorff dimension of the set of closed orbits for a cylindrical transformation, Nonlinearity, 23, 2393–2422.

• Crossref
• Export Citation
• 

W.H. Gottschalk, G.A. Hedlund, (1955), Topological Dynamics, Amer. Math. Soc. Colloq. Publ. 36, Amer. Math. Soc., Providence, RI, 1–148.

• 

A. Ya. Khinchin. (1964), Continued fractions, The University of Chicago Press, Chicago – London (transl. from Russian).

• 

A. Kochergin, (2002), A mixing special flow over a circle rotation with almost Lipschitz function, Sbornik: Mathematics, 193, 359–385.

• Crossref
• Export Citation
• 

A. Kochergin, (2015), A Besicovitch Cylindrical Transformation with Hölder Function, Electronic Research Announcements in Mathematical Sciences, 22, 87 – 91. S 1935 – 9179 AIMS.

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L.G. Schnirel’mann, (1930), An example of a plane transformation, Izv. Donskogo Politekhnich. Inst., 14, 64–74 (in Russian)

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