## 1 Besicovitch cascades

*T*

_{ρ}: 𝕋 → 𝕋 be an irrational circle rotation

*f*: 𝕋 → ℝ be a continuous function with zero mean. We consider a

*cylindrical transformation T*

_{ρ,f}: 𝕋 × ℝ → 𝕋 × ℝ with a cocycle

*f*:

*cylindrical cascade*.) The iterations of a point (

*x,y*) are described by

*f*(

^{r}*x*) is a Birkhoff sum

Cylindrical transformation was studed by H. Poincaré [14] 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, [13], [2], [15].

*f*is a

*coboundary*over

*T*

_{ρ}in the class of continuous functions, i.e., there exists

*F*∈

*C*(𝕋) such that

*x*, then the orbit of each point (

*x*

_{0},

*y*

_{0}) is contained within a closed invariant curve

*T*

_{ρ,f}is isomorphic to the rotation of the cylinder around the axis.

Later L.G. Shnirel’man [16] and A.S. Besicovitch [3] 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 [8] showed that *T*_{ρ,f}*is topologically transitive if and only if f is not a coboundary over T*_{ρ}*and has zero mean*.

*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, [1]). The Birkhoff sums for coboundaries have a convenient description

*T*

_{ρ}, there exists a continuous

*f*such that

*T*

_{ρ,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

*f*

^{r}(

*x*) → ∞ as

*r*→ ± ∞, and so each orbit is discrete.

*ρ*. Recall the basic notions. Every irrational

*ρ*∈ (0,1) may be represented as an infinite continued fraction

*k*

_{n}are called

*partial quotients*. The fraction

*p*

_{n}/

*q*

_{n}= [

*k*

_{1},...,

*k*

_{n}] is called the

*convergent*of continued fraction. It is known [9], that the convergents

*p*

_{n}/

*q*

_{n}are determined recursively:

*ρ*is called Diophantine if there exist

*C*> 0,

*θ*> 0 such that

*n*. Otherwise, it is called Liouville.

In 2010, K. Frączek and M. Lemańczyk [7] 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 *T*_{ρ}, *T*_{ρ} is uniquely ergodic with the only invariant Lebesgue measure, and, therefore, *B* has a null Lebesgue measure.

Obviously, if *f* has bounded variation, then *T*_{f} is not Besicovitch, because the sequence
*q*_{n} of *ρ*, as *‖ f*^{qn}*‖*_{C(𝕋)} ⩽ Var(*f*).

K. Frączek and M. Lemańczyk showed that the Besicovitch cascade with continuous function *f* can be constructed for any transitive *T*_{ρ}. For *ρ* satisfying some Diophantine condition, the *γ*-Hölder function *f* was obtained, so that *T*_{ρ,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 [7] 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 [5], 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 [11], [12], 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 [11], using angles with property
*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 T*_{ρ}*such that the cylindrical transformation T*_{ρ,f}*is Besicovitch, and the Hausdorff dimension of the Besicovitch set in the circle is greater than* 1 − *γ− ɛ*.

In [12], using angles with relatively slowly varying, but infinitely large partial quotients, it was managed to achieve inequality dim_{H}(*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.

*For any γ* ∈ (0,1) *and any ɛ* > 0, *there exists a γ-Hölder function f and a circle rotation T*_{ρ}*with bounded quotients such that the cylindrical transformation T*_{ρ,f}*is Besicovitch, and the Hausdorff dimension of the Besicovitch set in the circle is greater than* 1 − *γ*− *ɛ*.

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

Is 1 − *γ* the upper bound for dim_{H}(*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*.

[7] 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 [11], but the operation with rotations having bounded partial quotients required a slight modification and more subtle estimates.

*ρ*= [

*k*

_{1},...,

*k*

_{n},...] be an irrational number, and {

*p*

_{n}/

*q*

_{n}} be the sequence of convergents to

*ρ*. Let

*δ*

_{n}= |

*ρ*−

*p*

_{n}/

*q*

_{n}|. It is well known that 1

*γ*-Hölder cocycle

*f*as the sum of a series of Lipschitz functions

*f*

_{n}corresponding to fractions

*p*

_{n}/

*q*

_{n}:

*f*

_{n}is 1/

*q*

_{n}–periodical function defined by expression

*t*

_{n}and

*a*

_{n}are used. We assume, that for any number

*n*

*t*

_{n}shows, what part of the period is occupied by the support of

*f*

_{n}, and

*a*

_{n}> 0 is the Lipschitz constant for

*f*

_{n}.

**Note.** Since the addition of the coboundary to the function
*p*_{n}/*q*_{n}, and also estimate the partial sums
*n*_{0}. For simplicity, in this case we shift the numbering, assuming *n*_{0} = 1.

*D*⊂ 𝕋, each point of which has a discrete orbit:

*If a*_{n} ⩽ const(*t*_{n}*q*_{n})^{1−γ}*for 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 [12]. 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 [10].

The convergence of series (4) follows from the inequality
*q*_{n} grow not slower than exponentially.

The conditions of the next lemma are modified compared with those in the previous papers [11] and [12]. These changes are adapted to dealing with bounded partial quotients.

*If the sequences*{

*q*

_{n}}, {

*t*

_{n}}

*and*{

*a*

_{n}}

*satisfy the conditions*

*then for any x*∈

*D*

^{+}

*For x*∈

*D*−,

*the situation is inverse.*

**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
*n*-th term for
*r* = *q*_{n}/4 grows up to a level of the order *a*_{n}/*t*_{n}, and at a moment of the order of
*q*_{n+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* = *q*_{n+1}/2, the Birkhoff sum
*f*^{r} on the set
*f*_{n} 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

*f*

_{n}is a coboundary. For this let us define a 1/

*q*

_{n}–periodic function

*F*

_{n}: 𝕋 → ℝ such that

*n*, when

*ρ*=

*p*

_{n}/

*q*

_{n}+

*δ*

_{n}, we immediately have (11). For odd

*n*, when

*ρ*=

*p*

_{n}/

*q*

_{n}−

*δ*

_{n},

*L*

_{n}, by (3) and (8), the estimates

**3.**Since for any integer

*r*and any

*x*∈ 𝕋

**4.**Now let be

*r*∈ [0,

*q*

_{n+1}/4]

*f*

_{n}and the set

*x*∈ [1/(4

*t*

_{n}

*q*

_{n}),3/(4

*t*

_{n}

*q*

_{n})]. For even

*n*, and given that (

*q*

_{n+1}/4)

*δ*

_{n}< 1/(4

*q*

_{n}),

*t*

_{n}> 2 and

*f*

_{n}is non-negative, which implies (16) and non-decreasing Birkhoff sums

*r*in the indicated range.

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

*r*= ⎡

*q*

_{n}/4 ⎤. As above, we can assume that

*x*∈ [1/(4

*t*

_{n}

*q*

_{n}),3/(4

*t*

_{n}

*q*

_{n})], and

*x*+ (−1)

^{n}

*kδ*

_{n}, 0 ⩽

*k*⩽

*q*

_{n}/4} fill evenly a segment of length shorter than 1/(4

*mt*

_{n}

*q*

_{n}) in the segment

**5.**Now we can estimate Birkhoff sum

*f*

^{r}(

*x*) for

*r*∈ [

*q*

_{n}/4,

*q*

_{n+1}/4]:

*n*→ ∞, and therefore, for

*r*→ +∞.

The cases *r* → − ∞ and
*s* > 1 require a shift in the numbering of terms.

Lemma 3 is proved.

## 3 Rotation of the circle

*ρ*with constant, starting from some position, partial quotients

*Q*is a rational number. In the standard representation with integer partial quotients, this means that partial quotients

*k*

_{n}=

*K*for all numbers

*n*large enough. Thus, we define the sequence {

*q*

_{n}} (and also {

*t*

_{n}} and {

*a*

_{n}}). After that we fix the Hölder constant

*γ*∈ (0,1) and the constant

*f*. It will give us, by (8),

*T*

_{ρ,f}, we define

*K*,

*γ*and

*m*, and we substitute them to the conditions of Lemma 3.

*n*,

*q*

_{n}} satisfies the difference equation, for

*n*large enough we have

*λ*

^{2}−

*λ*− 1 = 0. Also we have

*a*

_{n}≈

*b*

_{n}if

*a*

_{n}=

*b*

_{n}(1 +

*O*(1/

*λ*

^{−2n})) for

*n*→ ∞. In this notation

*a*

_{n}= (

*t*

_{n}

*q*

_{n})

^{1}−

^{γ}, we have

*If the parameters γ, m, K are such that*

*then the conditions of Lemma 3 are satisfied.*

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

## 4 Hausdorff dimension of Besicovitch Set

*B*, i. e., set of points on the circle 𝕋 × ℝ having discrete orbits. In section 2, the subset

*D*⊂

*B*of Besicovitch set was constructed (see (6)):

_{H}

*D*is also for dim

_{H}

*B*. As for the upper bound for dim

_{H}

*D*, the paper does not prove that it is such for dim

_{H}

*B*, but such a hypothesis exists.

*D*is the union of a countable set of sets of Cantor type. It is sufficient to estimate

*The Hausdorff dimension of Besicovitch subset D satisfies*

*As a result,*

Recall, that
*n*-th level segments.

_{H}

*D*by definition (see [6], section 2.1). Define the covers

*W*

_{n}of

*W*

_{n}consists of those segments of

*W*

_{n}

*−*

_{1}in more than one point. The length of each of them is equal to 1/(2

*t*

_{n}

*q*

_{n}). By virtue of (8), each

*n*th-level segment may intersect

*m*/2 or

*m*/2 + 1 segments of (

*n*+ 1)-th level. If

*m*

_{n}is the number of

*n*th–level segments included into

*W*

_{n}, than

*s*

_{n}the minimal number of

*n*th-level segments, contained in one (

*n*− 1)th-level segment. Let

*ɛ*

_{n}be the minimal distance between

*n*-th level segments. Then

*n*-th level segment contains entirely

*m*/2 or (

*m*/2 − 1) segments of the form [(

*i*− 1) /

*q*

_{n+1},

*i/q*

_{n+1}], therefore, each

*n*th-level segment entirely contains at least (

*m*/2 − 1) (

*n*+ 1)-th level segments, where from

*s*

_{n}⩾

*m*/2 − 1, we have

*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*

*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 {*p*_{n}/*q*_{n}}. Now, according to (8), the sequences {*t*_{n}} and {*a*_{n} = (*t*_{n}*q*_{n})^{1}−^{γ}} 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 *x* ∈ *D* run away to infinity under iterations of the cylindrical cascade *T*_{ρ,f}. So *T*_{ρ,f} is a Besicovitch cascade, and *D* is a Besicovitch subset. By Lemma 2, *f* is *γ*-Hölder.

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