Hausdorff dimensions of perturbations of a conformal iterated function system via thermodynamic formalism

Abstract

We consider small perturbations of a conformal iterated function system produced by either adding or removing some generators with small derivative from the original. We establish a formula, utilizing transfer operators arising from the thermodynamic formalism à la Sinai–Ruelle–Bowen, which may be solved to express the Hausdorff dimension of the perturbed limit set in series form: either exactly, or as an asymptotic expansion. Significant applications to the dimension theory of continued fraction Cantor sets include strengthening Hensley’s asymptotic formula from 1992, which improved on earlier bounds due to Jarník and Kurzweil, for the Hausdorff dimension of the set of real numbers whose continued fraction expansion partial quotients are all $\le N$; as well as its counterpart for reals whose partial quotients are all $\ge N$ due to Good from 1941.

Appendices

Appendix A: Computation of coefficients

Although we do not state precise formulas for the coefficients $c_{I,j,k}$ and $c_{I,j}$ appearing in Theorem 4.1, our proofs do facilitate the construction of such formulas. Namely, Lemmas 6.2 and 6.3 allow us to write $\alpha$ and $\beta$ in terms of the secondary operators $\alpha _j$ and $\beta _{i,j}$. Plugging these into (2.4) gives the coefficients for the power series $\Xi$ in terms of $(\eta _i)$, $\theta$, and $\xi$.

To illustrate this process, we compute Hensley’s coefficients $c_{1,0}$ and $c_{2,1}$ for the sequence of systems $E_N = \{1,\ldots ,N\} \rightarrow E = \mathbb {N}$, and we show that the formula for $c_{2,0}$ in (1.3) involves Apery’s constant $\zeta (3)$ as well as the expressions

$$\begin{aligned} \mu M_\phi&Q L M_\phi g,&\mu M_\phi&Q h,&\nu&Q L M_\phi g,&\nu&Q h, \end{aligned}$$

(A.1)

where the notation is as in §9, and $M_\phi$ denotes multiplication by the function

$$\begin{aligned} \phi (x) = 2\log (x). \end{aligned}$$

Note that it appears to be impossible to rewrite even the simplest of these expressions, $\nu Q h = Q\mathbbm {1}(0) = 1 + \sum _{n\ge 1} (L^n \mathbbm {1}(0) - 1/\log (2))$, in closed form. However, all of the expressions can be approximated with arbitrary accuracy.

To this end, in what follows we let $X = N^{-1} \log (N)$. Note that since $\theta = O(N^{-1})$, we have $c_{I,j} \, \eta _I \theta ^j \equiv _{X^k} 0$ whenever $\#(I) + \Sigma (I) + j \ge k$. In particular, the second-order approximation of (2.4) is

$$\begin{aligned} \mu \alpha g + \mu \beta g + \mu \Delta Q \Delta g \underset{X^3}{\equiv }\ \Xi = 0. \end{aligned}$$

Next we observe that $\phi \circ u_b(x) = \log |u_b'(x)|$ for all b, x. Thus since $\delta = 1$, for all $j \ge 1$ we have

$$\begin{aligned} \alpha _j f(x)= & {} \frac{1}{j!} \sum _{b \in S} |u_b'(x)| \log ^j|u_b'(x)| \; f\circ u_b(x) = \frac{1}{j!} \sum _{b\in S} |u_b'(x)| (\phi ^j f)\circ u_b(x) \\= & {} \frac{1}{j!} L M_\phi ^j f(x) \end{aligned}$$

so

$$\begin{aligned} \mu \alpha _j g&= \frac{1}{j!} \mu L M_\phi ^j g = \frac{1}{j!} \mu M_\phi ^j g\\&= \frac{2^j}{j!} \int _0^1 \frac{\log ^j(x)}{1 + x} \;\textrm{d}x = \frac{2^j}{j!} \sum _{n = 0}^\infty (-1)^n \int _0^1 x^n \log ^j(x) \;\textrm{d}x \\&= 2^j \sum _{n = 0}^\infty (-1)^n \sum _{j' = 0}^j \frac{(-1)^{j'}}{(n+1)^{j' + 1} (j - j')!} \big [x^{n+1} \log ^{j - j'}(x)\big ]_{x=0}^1 \\&= (-1)^j 2^j \sum _{n = 0}^\infty \frac{(-1)^n}{(n+1)^{j+1}} = (-1)^j 2^j (1 - 2^{-j}) \zeta (j+1)\\&= (-1)^j (2^j - 1) \zeta (j + 1) \end{aligned}$$

and thus

$$\begin{aligned} \mu \alpha g&= \sum _{j=1}^\infty \theta ^j \mu \alpha _j g \underset{X^3}{\equiv }\ \theta \mu \alpha _1 g + \theta ^2 \mu \alpha _2 g = -\zeta (2)\theta + 3 \zeta (3) \theta ^2. \end{aligned}$$

On the other hand, by direct computation¹¹ we have

$$\begin{aligned} \mu \beta _{0,0} g&= 1,&\mu \beta _{0,1} g&= 0,&\mu \beta _{1,0} g&= \int _0^1 -2x-1 \;\textrm{d}x = -2 \end{aligned}$$

and

$$\begin{aligned} \eta _0&\underset{X^3}{\equiv }\ -\left( \frac{N^{-(1 + 2\theta )}}{1 + 2\theta } - \frac{N^{-(2 + 2\theta )}}{2}\right) \underset{X^3}{\equiv }\ -\frac{1}{N} + \frac{2\theta \log (N)}{N} + \frac{2\theta }{N} + \frac{1}{2N^2} \underset{X^2}{\equiv }\ -\frac{1}{N},\\ \eta _1&\underset{X^3}{\equiv }\ -\frac{N^{-(2 + 2\theta )}}{2 + 2\theta } \underset{X^3}{\equiv }\ -\frac{1}{2N^2} \end{aligned}$$

so

$$\begin{aligned} \mu \beta g&\underset{X^3}{\equiv }\ \eta _0 (\mu \beta _{0,0} g + \theta \mu \beta _{0,1} g) + \eta _1 (\mu \beta _{1,0} g)\\&\underset{X^3}{\equiv }\ \left( -\frac{1}{N} + \frac{2\theta \log (N)}{N} + \frac{2\theta }{N} + \frac{1}{2N^2}\right) (1) + \left( -\frac{1}{2N^2}\right) (-2) \end{aligned}$$

Finally, since $\beta _{0,0} = h \nu$, $\mu h = \nu g = 1$, and $\mu L = L$, we have

$$\begin{aligned} \mu \Delta Q \Delta g&\underset{X^3}{\equiv }\ (\theta \mu M_\phi + \eta _0 \nu ) Q (\theta L M_\phi g + \eta _0 h). \end{aligned}$$

Next we compute the first-order approximation of $\Xi$:

$$\begin{aligned} \Xi \underset{X^2}{\equiv }\ -\zeta (2)\theta + (-1/N) \end{aligned}$$

and so setting $\Xi = 0$ yields $\theta \equiv _{X^2} -1/\zeta (2)N$, giving Hensley’s first coefficient

$$\begin{aligned} c_{1,0} = -1/\zeta (2) = -6/\pi ^2. \end{aligned}$$

Plugging this into the above formulas gives

$$\begin{aligned} \theta =&\frac{1}{\zeta (2)}\big (\Xi + \zeta (2)\theta \big ) \underset{X^3}{\equiv }\ \frac{1}{\zeta (2)} \left[ - \frac{1}{N} - \frac{2}{\zeta (2)} \frac{\log (N)}{N^2} + \left( \frac{3}{2} - \frac{2}{\zeta (2)} + \frac{3\zeta (3)}{\zeta ^2(2)}\right) \frac{1}{N^2}\right. \\&+ \left. \left( \frac{1}{\zeta (2)} \mu M_\phi + \nu \right) Q \left( \frac{1}{\zeta (2)} L M_\phi g + h\right) \frac{1}{N^2}\right] . \end{aligned}$$

This formula gives Hensley’s second coefficient $c_{2,1} = -2/\zeta ^2(2) = -72/\pi ^4$: the next coefficient is

$$\begin{aligned} c_{2,0} = \frac{3}{2} - \frac{2}{\zeta (2)} + \frac{3\zeta (3)}{\zeta ^2(2)} + \left( \frac{1}{\zeta (2)} \mu M_\phi + \nu \right) Q \left( \frac{1}{\zeta (2)} L M_\phi g + h\right) \end{aligned}$$

(A.2)

Notice that the four terms of (A.1) all appear in this formula.

1.1 A.1. Some further coefficients

It turns out to be possible to compute the coefficients $c_{i,i - 1}$ directly without dealing with any coefficients $c_{i,j}$ such that $j \le i - 2$. Namely, let us write $A \equiv _p B$ if

$$\begin{aligned} B - A \equiv N^{-p} \sum _{i = 0}^{\rightarrow \infty } \sum _{j = 0}^i c_{i,j} \frac{\log ^j(N)}{N^i} \end{aligned}$$

for some coefficients $c_{i,j}$. We can think of this as saying that $B - A$ is “formally $O(N^{-p})$”, in a sense where $N^{-1} \log (N)$ is considered “small” but $N^{-1} \log ^2(N)$ is not considered “small”.

Now (2.4) becomes

$$\begin{aligned} \mu \alpha g + \mu \beta g \underset{2}{\equiv }\ \Xi = 0. \end{aligned}$$

Moreover, similarly to before we have $\mu \alpha g \equiv _2 \theta \mu \alpha _1 g = -\zeta (2)\theta$. On the other hand,

$$\begin{aligned} \mu \beta g \underset{2}{\equiv }\ \eta _0 \mu \beta _{0,0} g = \eta _0 \underset{2}{\equiv }\ -N^{-1 - 2\theta } \end{aligned}$$

by the Euler-Maclaurin formula. Thus, (2.4) becomes

$$\begin{aligned} \zeta (2)\theta \underset{2}{\equiv }\ -N^{-(1 + 2\theta )} = -N^{-1} \exp (-2\theta \log (N)). \end{aligned}$$

So we have $-\zeta (2) N\theta \equiv _2 F(2\log (N)/\zeta (2)N)$, where

$$\begin{aligned} F(x) = \sum _{j = 0}^\infty a_j x^j \;\;\;\; \text { satisfies} \;\;\;\; F(x) = \exp (x F(x)), \end{aligned}$$

(A.3)

i.e. F is the inverse of $y\mapsto \log (y)/y$ defined in a neighborhood of 0 and sending 0 to 1. It follows that $c_{i,i - 1} = -(2^{i-1}/\zeta ^i(2)) a_{i-1}$.

To compute $a_j$, we first recall Cayley’s formula: the number of spanning trees on i points is $T_i = i^{i - 2}$. To produce a recursive formula for $(T_i)$, observe that to define a spanning tree on i points, you need to define (a) a partition of the set of $i-1$ points, (b) spanning trees on each element of the partition, and (c) a root node in each of these spanning trees to connect to the final node to form the overall tree.

Now compare the recursive formulas for $(a_j)$ and $(T_i)$:

$$\begin{aligned} a_j&= \sum _{n = 0}^\infty \frac{1}{n!} \sum _{\begin{array}{c} t\in \mathbb {N}^n \\ |t| + n = j \end{array}} \prod _{k = 1}^n a_{t_k} = \sum _{P\in \mathcal {P}_j} \prod _{A\in P} \#(A)! a_{\#(A) - 1}\\ T_i&= \sum _{P\in \mathcal {P}_{i - 1}} \prod _{A\in P} \#(A) T_{\#(A)} \end{aligned}$$

where $\mathcal {P}_n$ is the set of all partitions of $\{1,\ldots ,n\}$. It follows that

$$\begin{aligned} a_j = \frac{T_{j + 1}}{j!} = \frac{(j + 1)^{j - 1}}{j!} \cdot \end{aligned}$$

So

$$\begin{aligned} c_{i,i-1} = -\frac{2^{i-1}}{\zeta ^i(2)} \frac{i^{i-2}}{(i-1)!}, \end{aligned}$$

which is equivalent to (1.4) from the introduction.

Appendix B: Definition of a conformal iterated function system (CIFS)

We recall the definition of a conformal iterated function system (CIFS) due to Mauldin–Urbański.

Definition B.1

(Cf. [68, p.108–110]). Fix $d\in \mathbb {N}$. A collection of maps $(u_a)_{a\in E}$ is called a conformal iterated function system (CIFS) on $\mathbb {R}^d$ if:

• 1. E is a countable (finite or infinite) index set;

• 2. $X\subseteq \mathbb {R}^d$ is a nonempty compact set which is equal to the closure of its interior;

• 3. For all $a\in E$, $u_a(X) \subseteq X$;

• 4. (Cone condition)

  $$\begin{aligned} \inf _{\textbf{x}\in X, r\in (0,1)} \frac{\lambda (X\cap B(\textbf{x},r))}{r^d} > 0, \end{aligned}$$

  where $\lambda$ denotes Lebesgue measure on $\mathbb {R}^d$;

• 5. $V\subseteq \mathbb {R}^d$ is an open connected bounded set such that $d(X,\mathbb {R}^d\setminus V) > 0$;

• 6. For each $a\in E$, $u_a$ is a conformal homeomorphism from V to an open subset of V;

• 7. (Uniform contraction) $\sup _{a\in E} \sup |u_a'| < 1$, and if E is infinite, $\lim _{a\in E} \sup |u_a'| = 0$;

• 8. (Bounded distortion property) For all $n\in \mathbb {N}$, $\omega \in E^n$, and $\textbf{x},\textbf{y}\in V$,

  $$\begin{aligned} |u_\omega '(\textbf{x})| \asymp _\times |u_\omega '(\textbf{y})|, \end{aligned}$$

  (B.1)

  where

  $$\begin{aligned} u_\omega = u_{\omega _1}\circ \cdots \circ u_{\omega _n}. \end{aligned}$$

The CIFS is called an OSC CIFS if in addition it satisfies the open set condition (OSC), i.e. if the collection $(u_a({\text {Int}}(X)))_{a\in E}$ is disjoint.