Besicovitch cascades and bounded partial quotients

Let 𝕋 = ℝ/ℤ be a circle of length 1, T_ρ : 𝕋 → 𝕋 be an irrational circle rotation Tρx=x+ρ (mod1),{T_\rho }x = x + \rho\,\,\,\, (\bmod 1), and let 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 : Tρ,f(x,y)=(Tρx,y+f(x)),{T_{\rho ,f}}(x,y) = ({T_\rho }x,y + f(x)), which is a skew product over the circle rotation. (D.V. Anosov [1] 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,T_{\rho ,f}^{r}(x,y)=(T_{\rho }^{r}x,y+{{f}^{r}}(x)),\quad r\in \mathbb{Z}, where f^r (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.f ^{r}(x)=\begin{cases}f(x)+f(T_{\rho}x)+\dots+f(T_{\rho}^{r-1}x) &\text{for } r>0,\\0\quad &\text{for } r=0,\\ -f(T_{\rho}^{-1}x)-\dots - f (T_{\rho}^{r+1}x)- f (T_{\rho}^{r}x) &\text{for } r<0.\end{cases}

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].

It turned out that there are cascades having unbounded orbits, and there are those only with bounded ones. If f is a coboundary over T_ρ in the class of continuous functions, i.e., there exists F ∈ C(𝕋) such that (1)f(x)=F(Tρx)−F(x)f(x) = F({T_\rho }x) - 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},\left\{ (x,y):y={{y}_{0}}+F(x)-F({{x}_{0}}),\quad x\in \mathbb{T} \right\}, and therefore it is bounded. In this case, the cylinder 𝕋 × ℝ splits into such curves, and 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,fis topologically transitive if and only if f is not a coboundary over T_ρand 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, [1]). The Birkhoff sums for coboundaries have a convenient description (2)fr(x)=F(Tρrx)−F(x).{f^r}(x) = F(T_\rho ^rx) - F(x).

In 1951 A.S. Besicovitch [4] proved that a cylindrical cascade cannot be minimal, it has nontransitive orbits, and for any irrational circle rotation 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 lim|r|→+∞fr(x)=∞.\mathop {\lim }\limits_{|r| \to + \infty } {f^r}(x) = \infty . The condition ∫Tf(x) dx=0\int_{\mathbb{T}}{}f(x)dx=0 is essential. Otherwise, by ergodic theorem, f^r(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+⋯,\rho = {1 \over {{k_1} + {1 \over {{k_2} + \cdots }}}}, or briefly, ρ=[k1,k2,…],\rho = [{k_1},{k_2}, \ldots ], and the natural numbers k_n are called partial quotients. The fraction p_n/q_n = [k₁,..., k_n] is called the convergent of continued fraction. It is known [9], that the convergents p_n/q_n 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.\matrix{ {{p_{n + 1}} = {k_{n + 1}}{p_n} + {p_{n - 1}},\;n \ge 1,\;{p_0} = 0,\;{p_1} = 1,} \hfill \cr {{q_{n + 1}} = {k_{n + 1}}{q_n} + {q_{n - 1}},\;n \ge 1,\;{q_0} = 1,\;{q_1} = {k_1}.} \hfill \cr } ρ is called Diophantine if there exist C > 0, θ > 0 such that qn+1<Cqn1+θ{q_{n + 1}} < Cq_n^{1 + \theta } for any 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 Tρ,fqn(x,y)T_{\rho ,f}^{{q_n}}(x,y) is bounded for denominators 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 qn+1≍qn1+θ{q_{n + 1}} \asymp q_n^{1 + \theta } , 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_ρsuch that the cylindrical transformation T_ρ,fis 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.

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

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

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.

So, let ρ = [k₁,...,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 (3)1qn(qn+1+qn)<δn<1qnqn+1{1 \over {{q_n}({q_{n + 1}} + {q_n})}} < {\delta _n} < {1 \over {{q_n}{q_{n + 1}}}} and ρ=pnqn+(−1)nδn.\rho = {{{p_n}} \over {{q_n}}} + {( - 1)^n}{\delta _n}.

We will construct the γ-Hölder cocycle f as the sum of a series of Lipschitz functions f_n corresponding to fractions p_n/q_n: (4)f=∑n=1∞fn,f = \sum\limits_{n = 1}^\infty {f_n}, where f_n is 1/q_n–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].f_n(x)=\begin{cases}\frac{a_n}{2\delta_n}(x+\delta_n)^{2}, & x\in [-\delta_n,0],\\\frac{a_n\delta_n}{2}+a_nx, & x\in \left[0, \frac{1}{2t_nq_n}-\delta_n\right],\\\frac{a_n}{2t_nq_n}-\frac{a_n\delta_n}{4}-\frac{a_n}{\delta_n}\left( x-\frac{1}{2t_nq_n}+\frac{\delta_n}{2}\right)^{2},& x\in \left[ \frac{1}{2t_nq_n}-\delta_n, \frac{1}{2t_nq_n}\right],\\f_n\left( \frac{1}{t_nq_n}-\delta_n-x\right),& x\in \left[ \frac{1}{2t_nq_n}, \frac{1}{t_nq_n}\right],\\0, & x\in \left[ \frac{1}{t_nq_n}, \frac{1}{2q_n}-\delta_n\right],\\-f_n\left(x-\dfrac{1}{2q_n}\right), & x\in \left[ \frac{1}{2q_n}-\delta_n, \frac{1}{q_n}-\delta_n\right].\end{cases} In this expression, in addition to the rotation number parameters, the quantities t_n and a_n are used. We assume, that for any number n(5)tn>2.{t_n} > 2. 1/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.

Fig. 1

Note. Since the addition of the coboundary to the function f=∑i=1∞fif = \sum\limits_{i = 1}^\infty {f_i} does not change the Besikovich set, we can consider the terms and the convergents p_n/q_n, and also estimate the partial sums ∑i=n0nfir\sum\limits_{i = {n_0}}^n f_i^r , 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: (6)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−.\matrix{ {G_n^ + } \hfill & { = \bigcup\limits_{k = 0}^{{q_n} - 1} \left( {{k \over {{q_n}}} + \left[ {{1 \over {4{t_n}{q_n}}},{3 \over {4{t_n}{q_n}}}} \right]} \right),\quad \quad G_n^ - = {1 \over {2{q_n}}} + G_n^ + ,} \hfill \cr {D_s^ + } \hfill & { = \bigcap\limits_{n = s}^\infty G_n^ + ,\quad \quad D_s^ - = \bigcap\limits_{n = s}^\infty G_n^ - ,} \hfill \cr {{D^ + }} \hfill & { = \bigcup\limits_{s = 1}^\infty D_s^ + ,\quad \quad {D^ - } = \bigcup\limits_{s = 1}^\infty D_s^ - ,} \hfill \cr D \hfill & { = {D^ + } \cup {D^ - }.} \hfill \cr }

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 ∥fn∥C⩽const(tnqn)γ\parallel {f_n}{\parallel _C} \leqslant {{\rm const} \over {{{({t_n}{q_n})}^\gamma }}} and the fact that denominators q_n grow not slower than exponentially.

In this article we set (7)an=(tnqn)1−γ.{a_n} = ({t_n}{q_n}{)^{1 - \gamma }}.

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.

We consider an irrational number ρ with constant, starting from some position, partial quotients ρ=[Q,K,K,…],\rho = [Q,K,K, \ldots ], where 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 m⩾6m\geqslant 6 appearing in the construction of f. It will give us, by (8), tn=qn+1mqn{t_n} = {{{q_{n + 1}}} \over {m{q_n}}} . To completely determine the cylindrical cascade T_ρ,f, we define (19)an=(tnqn)1−γ.{a_n} = {\left( {{t_n}{q_n}} \right)^{1 - \gamma }}. 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.{p_{n + 1}} = K{p_n} + {p_{n - 1}},\quad {q_{n + 1}} = K{q_n} + {q_{n - 1}}. Since the sequence {q_n} satisfies the difference equation, for n large enough we have qn=C1λn+C2(−λ)−n,{q_n} = {C_1}{\lambda ^n} + {C_2}{( - \lambda )^{ - n}}, where λ=K2+K24+1≈K+1K\lambda = {K \over 2} + \sqrt {{{{K^2}} \over 4} + 1} \approx K + {1 \over K} is the root of characteristic equation λ² − λ − 1 = 0. Also we have [K,K,K,…]=1λ, ρ=[Q,K,K,…]=1Q+1/λ.[K,K,K, \ldots ] = {1 \over \lambda },\quad \rho = [Q,K,K, \ldots ] = {1 \over {Q + 1/\lambda }}.

We say that a_n ≈ b_n if a_n = b_n (1 + O(1/λ⁻²ⁿ)) for n → ∞. In this notation (20)qn≈Cλn, qn+1qn≈λ.{q_n} \approx C{\lambda ^n},\quad \quad {{{q_{n + 1}}} \over {{q_n}}} \approx \lambda .

Using equalities qn+1tnqn=m{{{q_{n + 1}}} \over {{t_n}{q_n}}} = m (condition (8)) and a_n = (t_nq_n)¹−^γ, we have (21)tn=qn+1mqn≈λm,{t_n} = {{{q_{n + 1}}} \over {m{q_n}}} \approx {\lambda \over m},(22)an=(tnqn)1−γ≈(Cλm)1−γλn(1−γ),{a_n} = ({t_n}{q_n}{)^{1 - \gamma }} \approx {\left( {{{C\lambda } \over m}} \right)^{1 - \gamma }}{\lambda ^{n(1 - \gamma )}},(23)antn≈C1−γ(mλ)γλn(1−γ),{{{a_n}} \over {{t_n}}} \approx {C^{1 - \gamma }}{\left( {{m \over \lambda }} \right)^\gamma }{\lambda ^{n(1 - \gamma )}},(24)anqn+1tn2qn:an+1tn+1=qn+1tnqn⋅(antn:an+1tn+1)≈mλ1−γ.{{{a_n}{q_{n + 1}}} \over {t_n^2{q_n}}}:{{{a_{n + 1}}} \over {{t_{n + 1}}}} = {{{q_{n + 1}}} \over {{t_n}{q_n}}} \cdot \left( {{{{a_n}} \over {{t_n}}}:{{{a_{n + 1}}} \over {{t_{n + 1}}}}} \right) \approx {m \over {{\lambda ^{1 - \gamma }}}}.

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 D ⊂ B of Besicovitch set was constructed (see (6)): limr→±∞fr(x)=∞ for any x∈D.\mathop {\lim }\limits_{r \to \pm \infty } {f^r}(x) = \infty \quad {\rm{for\,any}}\,x \in D. Thus, the lower bound for dim_HD is also for dim_HB. As for the upper bound for dim_HD, the paper does not prove that it is such for dim_HB, 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+)\mathop {\dim }\nolimits_H (D_1^ + ) because dimH(D)=dimH(D1+).\mathop {\dim }\nolimits_H (D) = \mathop {\dim }\nolimits_H (D_1^ + ).

Recall, that D1+=∩n=1∞Gn+D_1^ + = \bigcap\limits_{n = 1}^\infty G_n^ + (see (6)). Let us call the segments of Gn+G_n^ + as n-th level segments.

We get the upper bound for dim_HD by definition (see [6], section 2.1). Define the covers W_n of D1+D_1^ + by segments inductively. W1=G1+{W_1} = G_1^ + (we consider the cover as the union of segments). The cover W_n consists of those segments of Gn+G_n^ + that intersect W_n−₁ in more than one point. The length of each of them is equal to 1/(2t_nq_n). By virtue of (8), each nth-level segment may intersect m/2 or m/2 + 1 segments of (n + 1)-th level. If m_n is the number of nth–level segments included into W_n, than mn⩽(m2+1)mn−1{m_n} \leqslant \left( {{m \over 2} + 1} \right){m_{n - 1}}, , or mn⩽m1(m2+1)n−1.{m_n} \leqslant {m_1}{\left( {{m \over 2} + 1} \right)^{n - 1}}. Thus, we have the cover of D1+D_1^ + by segments of length 1/(2tnqn)≈Cm⋅λn+11/(2{t_n}{q_n}) \approx {C \over m} \cdot {\lambda ^{n + 1}} (by (20) and (5)). Then we have the estimate Hnd=mn⋅1/(2tnqn)d<const(m2+1λd)n−1.H_n^d = {m_n} \cdot 1/(2{t_n}{q_n}{)^d} < {\rm const}{\left( {{{{m \over 2} + 1} \over {{\lambda ^d}}}} \right)^{n - 1}}. If λd>(m2+1){\lambda ^d} > \left( {{m \over 2} + 1} \right) , then limn→∞Hnd=0\mathop {\lim }\limits_{n \to \infty } H_n^d = 0 . Therefore, for Hausdorff dimension d=dimHDn+d = \mathop {\dim }\nolimits_H D_n^ + , the inequality λd⩽(m2+1){\lambda ^d} \leqslant \left( {{m \over 2} + 1} \right) is necessary, and we get the upper bound for Hausdorff dimension.

We obtain the lower bound for Hausdorff dimension by the method proposed in [6], Example 4.6. Now we denote by s_n the minimal number of nth-level segments, contained in one (n − 1)th-level segment. Let ɛ_n be the minimal distance between n-th level segments. Then (28)dimH(D1+)⩾liminfn→∞ln(s2…sn)−ln(sn+1εn+1).\mathop {\dim }\nolimits_H (D_1^ + )\geqslant \mathop {\liminf}\limits_{n \to \infty } {{\ln ({s_2} \ldots {s_n})} \over { - \ln ({s_{n + 1}}{\varepsilon _{n + 1}})}}. By (8), each 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 nth-level segment entirely contains at least (m/2 − 1) (n + 1)-th level segments, where from s_n ⩾ m/2 − 1, we have εn=1qn(1−12tn){\varepsilon _n} = {1 \over {{q_n}}}\left( {1 - {1 \over {2{t_n}}}} \right) . Substituting these inequalities and (20) to (29), we obtain (29)dimH(D1+)⩾limn→∞(n−1)ln(m2−1)lnqn+1+lnsn+1−ln(1−12tn)=ln(m2−1)lnλ,\mathop {\dim }\nolimits_H (D_1^ + )\geqslant \mathop {\lim }\limits_{n \to \infty } {{(n - 1)\ln \left( {{m \over 2} - 1} \right)} \over {\ln {q_{n + 1}} + \ln {s_{n + 1}} - \ln \left( {1 - {1 \over {2{t_n}}}} \right)}} = {{\ln \left( {{m \over 2} - 1} \right)} \over {\ln \lambda }}, and we get the lower bound in (26).

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_nq_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.

As m and K satisfy (25), from Lemma 5 we get the estimate dimHB>1−γ−ε.\mathop {\dim }\nolimits_H B > 1 - \gamma - \varepsilon .