function schemes that have attracted attention of other authors and give
estimates on the Hausdorff dimension of their limit sets.

####  4.9.1 Non-linear fractional example

So far we have been studying iterated function schemes generated by linear
fractional transformations. Following [41], §6 we will consider a simple
example of a map of $\mathbb{H}^{2}$ of a different nature. For any $0<t\leq
1$ we can define

$f_{t}(z)=\frac{z}{t}-\frac{1}{z}.$

If $t=1$ then the real line is $f$-invariant and there is no strictly smaller
closed invariant set. If $0<t<1$ then there there exist a $f$-invariant Cantor
$X_{t}\subset\mathbb{R}$ [49].

######  Example 4.36 ($t=\frac{1}{2}$).

The map $f(z)=2z-\frac{1}{z}$ has a limit set $X\subset[-1,1]$ and there are
two inverse branches $T_{1},T_{2}\colon[-1,1]\to[-1,1]$ given by

$\displaystyle T_{1}(x)$ $\displaystyle=\frac{1}{4}(x-\sqrt{8+x^{2}]})$ (54)
$\displaystyle T_{2}(x)$ $\displaystyle=\frac{1}{4}(x+\sqrt{8+x^{2}}),$ (55)

which define a Bernoulli system on $[-1,1]$. The transfer operator defined by
(1) takes the form

$(\mathcal{L}_{t}f)(x)=f(T_{1}(x))|T_{1}^{\prime}(x)|^{t}+f(T_{2}(x))|T_{2}^{\prime}(x)|^{t}.$

We may observe that $T_{1}(x)=-T_{2}(-x)$ and
$T_{1}^{\prime}(x)=T^{\prime}_{2}(-x)$. It follows that the transfer operator
preserves subspaces consisiting of odd and even functions. Applying the
bisection method with $S=[-1,1]$, $m=10$ we obtain that101010Table 14 in [41]
gives an estimate $0.49344815$, which is correct except for the last two
significant figures. An inconsequential typographical mistake in [41] is that
there is an incorrect sign in the equation in the caption to Table 14.

$\dim_{H}X=0.4934480908\,025\pm 5\cdot 10^{-13}.$

The corresponding test functions $f$ and $g$ for $t_{0}=0.493448088\,02$ and
$t_{1}=0.4934480908\,03$, respectively, turn out to be even and given by

$f(x)=\sum_{n=0}^{7}a_{2n}x^{2n}\qquad g(x)=\sum_{n=0}^{7}b_{2n}x^{2n},$

which are plotted in Figure 8 and whose coefficients are given in §A.4.

Figure 8: A plot of the function $f$ for the system (55) (the function $g$
being similar)

We can also compute

$\inf_{S}\frac{\mathcal{L}_{t_{0}}f}{f}>1+10^{-13}\qquad\sup_{S}\frac{\mathcal{L}_{t_{1}}g}{g}<1-10^{-13}.$

to justify the dimension estimates above.

######  Remark 4.37.

These estimates can easily be improved by increasing the number of Chebyshev
points. For example, letting $m=20$ gives a better estimate

$\dim_{H}(X)=0.4934480908\,02613\pm 10^{-15}.$

####  4.9.2 Hensley Examples

In a well known article from 1992, Hensley [19] presented an algorithm for
calculating the Hausdorff dimension of the limit sets for suitable iterated
function schemes. In this article there was included a table containing
estimates on selected examples which were good for the computational resources
available at the time and continued to be quoted up to the present time. It is
a simple matter to apply the method in §3 to improve these estimates. We give
lower and upper bounds in Table 3.

Alphabet $A$ | Hensley Estimate | $d$
---|---|---
$1,2$ | $0.5312805062\,772051416$ | $0.5312805062\,7720514162\,4$
$1,3$ | $0.4544827$ | $0.4544890776\,6182874384\,5$
$1,4$ | $0.4111827$ | $0.4111827247\,7479177684\,4$
$2,3$ | $0.337437$ | $0.3374367808\,0606363630\,4$
$2,4$ | $0.306313$ | $0.3063127680\,5278403027\,7$
$3,4$ | $0.263737$ | $0.2637374828\,9742655875\,0$
$1,2,3$ | $0.7056609080\,$ | $0.7056609080\,2873823060\,7$
$1,2,4$ | $0.66922149$ | $0.6692214869\,1028607643\,2$
$1,3,4$ | $0.6042422606\,9111965$ | $0.6042422577\,5648956551\,0$
$2,3,4$ | $0.480696$ | $0.4806962223\,1757304132\,2$
$1,2,3,4$ | $0.788946$ | $0.7889455574\,8315397254\,0$
$1,2,7$ | $0.6179036954\,6338$ | $0.6179036954\,6337565066\,2$
$1,3,7$ | $0.55324225$ | $0.5532422505\,6731096881\,6$
$1,4,7$ | $0.51788376$ | $0.5178837570\,0691696528\,4$
$2,3,7$ | $0.43801241$ | $0.4380124057\,1403118230\,1$
$2,4,7$ | $0.410329$ | $0.4103293158\,3768700408\,7$
$3,4,7$ | $0.36757914$ | $0.3675791395\,9190093176\,3$
$1,2,3,7$ | $0.75026306$ | $0.7502630613\,3714304325\,2$
$1,2,4,7$ | $0.7185418875\,$ | $0.7185418874\,7036994981\,8$
$2,3,4,7$ | $0.540036$ | $0.5400358121\,5475951902\,6$
$1,2,3,4,7$ | $0.820004$ | $0.8200039471\,2686900746\,5$
$10,11$ | $0.146921$ | $0.1469212353\,9078346331\,0$
$100,10000$ | $0.052247$ | $0.0522465926\,3865887865\,1$
$2,7$ | $0.26022398$ | $0.2602238774\,2217867170\,7$
$1,3,4,7$ | $0.66015538$ | $0.6601553798\,3237807776\,6$
$1,7$ | $0.34623824$ | $0.3462382435\,3395787983\,0$
$4,7$ | $0.2052533419\,4$ | $0.2052534193\,6736493221\,5$
$3,7$ | $0.2249239471\,918$ | $0.2249239471\,9177898918\,3$
$1,2,3,4,5$ | $0.8368294437\,$ | $0.8368294436\,8120882244\,2$
$2,3,4,5$ | $0.55963645$ | $0.5596364501\,6477671331\,0$
$2,3,5$ | $0.4616137$ | $0.4616136840\,1828922267\,4$
$1,500$ | $0.1094760117\,37$ | $0.1094760117\,3723275274\,5$
Table 3: Numerical data for Hensley examples from [19]; $\dim_{H}X_{A}=d\pm
10^{-20}$.

Alphabet $A$ | Jenkinson | $d$
---|---|---
Estimate
$1,3,8$ | $0.5438$ | $0.5438505824\,0696620129\,10871$
$1,3,6$ | $0.5652$ | $0.5652752192\,8623250537\,07768$
$1,3,5$ | $0.5813$ | $0.5813668211\,3469731449\,44763$
$1,2,10$ | $0.5951$ | $0.5951117365\,4560755184\,18957$
$1,3,4$ | $0.6042$ | $0.6042422576\,9541848140\,50596$
$1,2,7,40$ | $0.6265$ | $0.6265741168\,9229403866\,4271$
$1,2,5$ | $0.6460$ | $0.6460620828\,3482621991\,26074$
$1,2,5,40$ | $0.6532$ | $0.6532480771\,5774872727\,88226$
$1,2,4$ | $0.6692$ | $0.6692214868\,6131601289\,10582$
$1,2,4,40$ | $0.6754$ | $0.6754204446\,6970405658\,86491$
$1,2,4,15$ | $0.6899$ | $0.6899117699\,4923640369\,39765$
$1,2,4,6$ | $0.7275$ | $0.7275240485\,5844736070\,17215$
$1,2,4,5$ | $0.7400$ | $0.7400268606\,0207750663\,59866$
$1,2,3,6$ | $0.7588$ | $0.7588596765\,7522348478\,34758$
$1,2,3,5$ | $0.7709$ | $0.7709149398\,4418222560\,66922$
$1,2,3,4,10$ | $0.8081$ | $0.8081711218\,9508471948\,06225$
$1,2,3,4,6$ | $0.8269$ | $0.8269084945\,9163116837\,24267$
$1,2,3,4,5,9$ | $0.8541$ | $0.8541484705\,3932261542\,70362$
$1,2,3,4,5,7$ | $0.8616$ | $0.8616561744\,0626491056\,99743$
$1,2,3,4,5,6$ | $0.8676$ | $0.8676191730\,6718378091\,2243$
$1,2,3,4,5,6,8$ | $0.8851$ | $0.8851175915\,5644894823\,12343$
$1,2,3,4,5,6,7$ | $0.8889$ | $0.8889553164\,9195167843\,64394$
$1,2,\ldots,8$ | $0.9045$ | $0.9045526893\,2916142728\,20095$
$1,2,\ldots,9$ | $0.9164$ | $0.9164211122\,6835174040\,64645$
$1,2,\ldots,10$ | $0.9257$ | $0.9257375908\,8754612367\,25506$
$1,2,\ldots,13$ | $0.9445$ | $0.9445341091\,7126158776\,76671$
$1,2,\ldots,18$ | $0.961$ | $0.9611931848\,1599230516\,44346$
$1,2,\ldots,34$ | $0.980$ | $0.9804196247\,7958255969\,58015$

Table 4: Numerical data for Jenkinson examples from [24]; $\dim_{H}X_{A}=d\pm
2\cdot 10^{-24}$.

In Table 1 in another paper by Hensley [20], there are a number of numerical
results on Hausdorff dimension various of limit sets. Let us consider two
typical examples from the list.

##### (i).

Let $X_{1,2,7}=\left\\{[0;a_{1},a_{2},a_{3},\cdots]\mid
a_{n}\in\\{1,2,7\\}\right\\}$. Hensley presents an estimate

$\dim_{H}(X_{1,2,7})=0.6179036954\,6338,$

accurate to $13$ decimal places. The bisection method with $S=[0,1]$,
$\varepsilon=10^{-23}$ and $m=30$ gives

$\dim_{H}(X)=0.6179036954\,6337565066\,3413\pm 10^{-24},$

with the corresponding test functions $\underline{f}$ and $\underline{g}$
satisfying

$\inf_{S}\frac{\mathcal{L}_{t_{0}}g}{g}>1+10^{-24},\qquad\sup_{S}\frac{\mathcal{L}_{t_{1}}f}{f}<1-10^{-24}.$

##### (ii).

Let $X_{1,3,4}=\left\\{[0;a_{1},a_{2},a_{3},\cdots]\mid
a_{n}\in\\{1,3,4\\}\right\\}$. Hensley presents an estimate
$\dim_{H}(X_{1,3,5})=0.6042422606\,9111965$. However, this is only accurate to
seven decimal places (there seeming to be a typographical error) and applying
the bisection method with $S=[0,1]$, $\varepsilon=10^{-23}$ and $m=30$ we can
correct the estimate as follows.

$\dim_{H}(X)=0.6042422577\,5648956551\,0773\pm 10^{-24}.$

####  4.9.3 Other limit sets

In [42] Moreira considered a limit set $X$ for the IFS

$T_{1}(x)=\frac{1}{1+x}\mbox{ and }T_{2}(x)=\frac{1}{2+\frac{1}{2+x}}.$

After Theorem 3.4 therein he gives a rigorous estimate
$0.353<\dim_{H}(X)<0.3572$111111Their application was superseded by other
work, so the interest in this bound is mainly academic.. Applying the
bisection method with $\varepsilon=10^{-30}$ and $m=40$ we obtain

$\dim_{H}X=0.3554004768\,3384079791\,6306289490\,45\pm 5\cdot 10^{-32}.$

There are also additional examples studied by Jenkinson, in connection with
his numerical investigation of the Texan conjecture. It is a simple matter to
apply the bisection method to compute intervals $[t_{0},t_{1}]$ containing the
actual values and these are presented in Table 4.

##  Appendix A Coefficients for polynomials

For completeness, we collect together the coefficients of the polynomials
which appear in the proofs of the theorems. The exceptions to this is Theorem
1.5 where the polynomials are of degree $200$. However, in all of these
examples the reader may easily reconstruct these polynomials using the method
described in §3.2 and §3.3.

_Coefficients given in this section are exact rational numbers._

###  A.1 Estimates of $\dim_{H}\mathcal{M}\setminus\mathcal{L}$

####  A.1.1 Part 1:
$(\mathcal{M}\setminus\mathcal{L})\cap(\sqrt{5},\sqrt{13})$

We present coefficients of the test functions
$\underline{f}=(f_{1},f_{2},f_{3},f_{4})$ and

$\underline{g}=(g_{1},g_{2},g_{3},g_{4})$ used in (17).

$f_{j}=\sum_{k=0}^{7}a_{k}^{j}x^{k}\quad
g_{j}=\sum_{k=0}^{7}b_{k}^{j}x^{k},\qquad j=1,2,3,4.$

Straightforward calculation shows that the functions $f_{j}$ and $g_{j}$ are
monotone decreasing on $[0,1]$ and achieve their minima at $1$. For
convenience, we give a lower bound $s_{j}<\min(f_{j}(1),g_{j}(1))$.

$f_{1}$ | $f_{2}$ | $f_{3}$ | $f_{4}$
---|---|---|---
$a^{1}_{0}$ | $\phantom{-}0.9719420630$ | $a^{2}_{0}$ | $\phantom{-}0.6996881504$ | $a^{3}_{0}$ | $\phantom{-}0.5151068706$ | $a^{4}_{0}$ | $\phantom{-}1.0035153909$
$a^{1}_{1}$ | $-0.4083622154$ | $a^{2}_{1}$ | $-0.3257856180$ | $a^{3}_{1}$ | $-0.1503102708$ | $a^{4}_{1}$ | $-0.3675009146$
$a^{1}_{2}$ | $\phantom{-}0.2105627503$ | $a^{2}_{2}$ | $\phantom{-}0.1808409982$ | $a^{3}_{2}$ | $\phantom{-}0.0521402712$ | $a^{4}_{2}$ | $\phantom{-}0.1667939317$
$a^{1}_{3}$ | $-0.1166905024$ | $a^{2}_{3}$ | $-0.1054480473$ | $a^{3}_{3}$ | $-0.0190434708$ | $a^{4}_{3}$ | $-0.0824212096$
$a^{1}_{4}$ | $\phantom{-}0.0649256757$ | $a^{2}_{4}$ | $\phantom{-}0.0606297675$ | $a^{3}_{4}$ | $\phantom{-}0.0070462566$ | $a^{4}_{4}$ | $\phantom{-}0.0416454207$
$a^{1}_{5}$ | $-0.0321113965$ | $a^{2}_{5}$ | $-0.0305675512$ | $a^{3}_{5}$ | $-0.0024652813$ | $a^{4}_{5}$ | $-0.0191792606$
$a^{1}_{6}$ | $\phantom{-}0.0114531320$ | $a^{2}_{6}$ | $\phantom{-}0.0110148596$ | $a^{3}_{6}$ | $\phantom{-}0.0006866462$ | $a^{4}_{6}$ | $\phantom{-}0.0065321353$
$a^{1}_{7}$ | $-0.0020311821$ | $a^{2}_{7}$ | $-0.0019639179$ | $a^{3}_{7}$ | $-0.0001041463$ | $a^{4}_{7}$ | $-0.0011267545$
$s_{1}$ | $\phantom{-}0.6$ | $s_{2}$ | $\phantom{-}0.4$ | $s_{3}$ | $\phantom{-}0.4$ | $s_{4}$ | $\phantom{-}0.7$
$g_{1}$ | $g_{2}$ | $g_{3}$ | $g_{4}$
---|---|---|---
$b^{1}_{0}$ | $\phantom{-}0.9719420489$ | $b^{2}_{0}$ | $\phantom{-}0.6996881913$ | $b^{3}_{0}$ | $\phantom{-}0.5151068139$ | $b^{4}_{0}$ | $\phantom{-}1.0035153929$
$b^{1}_{1}$ | $-0.4083624405$ | $b^{2}_{1}$ | $-0.3257858144$ | $b^{3}_{1}$ | $-0.1503103363$ | $b^{4}_{1}$ | $-0.3675011259$
$b^{1}_{2}$ | $\phantom{-}0.2105629174$ | $b^{2}_{2}$ | $\phantom{-}0.1808411480$ | $b^{3}_{2}$ | $\phantom{-}0.0521403058$ | $b^{4}_{2}$ | $\phantom{-}0.1667940700$
$b^{1}_{3}$ | $-0.1166906125$ | $b^{2}_{3}$ | $-0.1054481493$ | $b^{3}_{3}$ | $-0.0190434862$ | $b^{4}_{3}$ | $-0.0824212916$
$b^{1}_{4}$ | $\phantom{-}0.0649257436$ | $b^{2}_{4}$ | $\phantom{-}0.0606298318$ | $b^{3}_{4}$ | $\phantom{-}0.0070462630$ | $b^{4}_{4}$ | $\phantom{-}0.0416454670$
$b^{1}_{5}$ | $-0.0321114321$ | $b^{2}_{5}$ | $-0.0305675854$ | $b^{3}_{5}$ | $-0.0024652837$ | $b^{4}_{5}$ | $-0.0191792835$
$b^{1}_{6}$ | $\phantom{-}0.0114531451$ | $b^{2}_{6}$ | $\phantom{-}0.0110148723$ | $b^{3}_{6}$ | $\phantom{-}0.0006866469$ | $b^{4}_{6}$ | $\phantom{-}0.0065321434$
$b^{1}_{7}$ | $-0.0020311845$ | $b^{2}_{7}$ | $-0.0019639202$ | $b^{3}_{7}$ | $-0.0001041464$ | $b^{4}_{7}$ | $-0.0011267559$
$s_{1}$ | $\phantom{-}0.6$ | $s_{2}$ | $\phantom{-}0.4$ | $s_{3}$ | $\phantom{-}0.4$ | $s_{4}$ | $\phantom{-}0.7$

####  A.1.2 Part 2: $(\mathcal{M}\setminus\mathcal{L})\cap(\sqrt{13},3.84)$

We present coefficients of the test functions
$\underline{f}=(f_{1},f_{2},f_{3})$ and $\underline{g}=(g_{1},g_{2},g_{3})$
used in (20).

$f_{j}=\sum_{k=0}^{7}a_{k}^{j}x^{k}\quad
g_{j}=\sum_{k=0}^{7}b_{k}^{j}x^{k},\quad j=1,2,3.$

Straightforward calculation shows that the functions $f_{j}$ and $g_{j}$ are
monotone decreasing on $[0,1]$ and achieve their minima at $1$. For
convenience, we give a lower bound $s_{j}<\min(f_{j}(1),g_{j}(1))$.

$f_{1}$ | $f_{2}$ | $f_{3}$
---|---|---
$a^{1}_{0}$ | $\phantom{-}0.8909247279$ | $a^{2}_{0}$ | $\phantom{-}1.0057862651$ | $a^{3}_{0}$ | $\phantom{-}0.4637543240$
$a^{1}_{1}$ | $-0.5666958388$ | $a^{2}_{1}$ | $-0.6057959903$ | $a^{3}_{1}$ | $-0.1989455013$
$a^{1}_{2}$ | $\phantom{-}0.3563184411$ | $a^{2}_{2}$ | $\phantom{-}0.3687733983$ | $a^{3}_{2}$ | $\phantom{-}0.0812067365$
$a^{1}_{3}$ | $-0.2268317558$ | $a^{2}_{3}$ | $-0.2307067719$ | $a^{3}_{3}$ | $-0.0328166697$
$a^{1}_{4}$ | $\phantom{-}0.1399500506$ | $a^{2}_{4}$ | $\phantom{-}0.1411336470$ | $a^{3}_{4}$ | $\phantom{-}0.0130694788$
$a^{1}_{5}$ | $-0.0741972949$ | $a^{2}_{5}$ | $-0.0745402261$ | $a^{3}_{5}$ | $-0.0048273012$
$a^{1}_{6}$ | $\phantom{-}0.0275558152$ | $a^{2}_{6}$ | $\phantom{-}0.0276379657$ | $a^{3}_{6}$ | $\phantom{-}0.0013941859$
$a^{1}_{7}$ | $-0.0049924560$ | $a^{2}_{7}$ | $-0.0050037264$ | $a^{3}_{7}$ | $-0.0002161328$
$s_{1}$ | $\phantom{-}0.5$ | $s_{2}$ | $\phantom{-}0.6$ | $s_{3}$ | $\phantom{-}0.3$
$g_{1}$ | $g_{2}$ | $g_{3}$
---|---|---
$b^{1}_{0}$ | $\phantom{-}0.8909246136$ | $b^{2}_{0}$ | $\phantom{-}1.0057862594$ | $b^{3}_{0}$ | $\phantom{-}0.4637545188$
$b^{1}_{1}$ | $-0.5666952424$ | $b^{2}_{1}$ | $-0.6057953968$ | $b^{3}_{1}$ | $-0.1989454076$
$b^{1}_{2}$ | $\phantom{-}0.3563178857$ | $b^{2}_{2}$ | $\phantom{-}0.3687728381$ | $b^{3}_{2}$ | $\phantom{-}0.0812066594$
$b^{1}_{3}$ | $-0.2268313254$ | $b^{2}_{3}$ | $-0.2307063388$ | $b^{3}_{3}$ | $-0.0328166280$
$b^{1}_{4}$ | $\phantom{-}0.1399497528$ | $b^{2}_{4}$ | $\phantom{-}0.1411333481$ | $b^{3}_{4}$ | $\phantom{-}0.0130694591$
$b^{1}_{5}$ | $-0.0741971263$ | $b^{2}_{5}$ | $-0.0745400571$ | $b^{3}_{5}$ | $-0.0048272932$
$b^{1}_{6}$ | $\phantom{-}0.0275557503$ | $b^{2}_{6}$ | $\phantom{-}0.0276379006$ | $b^{3}_{6}$ | $\phantom{-}0.0013941834$
$b^{1}_{7}$ | $-0.0049924440$ | $b^{2}_{7}$ | $-0.0050037144$ | $b^{3}_{7}$ | $-0.0002161324$
$s_{1}$ | $\phantom{-}0.5$ | $s_{2}$ | $\phantom{-}0.6$ | $s_{3}$ | $\phantom{-}0.3$

####  A.1.3 Part 3: $(\mathcal{M}\setminus\mathcal{L})\cap(3.84,3.92)$

We present coefficients of the polynomial test functions
$\underline{f}=(f_{1},\ldots,f_{7})$ and $\underline{g}=(g_{1},\ldots,g_{7})$
used in (23). It follows from the equality between columns of the transition
matrix $M$ that certain components are identical.

$\displaystyle f_{1}$ $\displaystyle=f_{2}=\sum_{k=0}^{7}a_{k}^{1}x^{k},\quad$
$\displaystyle g_{1}=g_{2}=\sum_{k=0}^{7}b_{k}^{1}x^{k};$ $\displaystyle
f_{4}$ $\displaystyle=f_{5}=f_{6}=\sum_{k=0}^{7}a_{k}^{4}x^{k},\quad$
$\displaystyle g_{4}=g_{5}=g_{6}=\sum_{k=0}^{7}b_{k}^{4}x^{k};$ $\displaystyle
f_{j}$ $\displaystyle=\sum_{k=0}^{7}a_{k}^{j}x^{k},\quad$ $\displaystyle
g_{j}=\sum_{k=0}^{7}b_{k}^{j}x^{k};\quad j=3,7,8,9.$

Straightforward calculation shows that the functions $f_{j}$ and $g_{j}$ are
monotone decreasing on $[0,1]$ and achieve their minima at $1$. For
convenience, we give a lower bound $s_{j}<\min(f_{j}(1),g_{j}(1))$.

$f_{1}$ | $f_{3}$ | $f_{4}$
---|---|---
$a^{1}_{0}$ | $\phantom{-}0.8752491446$ | $a^{3}_{0}$ | $\phantom{-}0.8242977486$ | $a^{4}_{0}$ | $\phantom{-}0.9435673129$
$a^{1}_{1}$ | $-0.5862938673$ | $a^{3}_{1}$ | $-0.5673413247$ | $a^{4}_{1}$ | $-0.6108763701$
$a^{1}_{2}$ | $\phantom{-}0.3817921234$ | $a^{3}_{2}$ | $\phantom{-}0.3753845810$ | $a^{4}_{2}$ | $\phantom{-}0.3898319394$
$a^{1}_{3}$ | $-0.2509102043$ | $a^{3}_{3}$ | $-0.2488176120$ | $a^{4}_{3}$ | $-0.2534505061$
$a^{1}_{4}$ | $\phantom{-}0.1594107381$ | $a^{3}_{4}$ | $\phantom{-}0.1587442151$ | $a^{4}_{4}$ | $\phantom{-}0.1601941335$
$a^{1}_{5}$ | $-0.0866513835$ | $a^{3}_{5}$ | $-0.0864513792$ | $a^{4}_{5}$ | $-0.0868796783$
$a^{1}_{6}$ | $\phantom{-}0.0328162711$ | $a^{3}_{6}$ | $\phantom{-}0.0327671074$ | $a^{4}_{6}$ | $\phantom{-}0.0328711315$
$a^{1}_{7}$ | $-0.0060331410$ | $a^{3}_{7}$ | $-0.0060262852$ | $a^{4}_{7}$ | $-0.0060406787$
$s_{1}$ | $\phantom{-}0.4$ | $s_{3}$ | $\phantom{-}0.4$ | $s_{4}$ | $\phantom{-}0.5$
$f_{7}$ | $f_{8}$ | $f_{9}$
---|---|---
$a^{7}_{0}$ | $\phantom{-}0.2183470684$ | $a^{8}_{0}$ | $\phantom{-}0.5232653503$ | $a^{9}_{0}$ | $\phantom{-}1.0058222185$
$a^{7}_{1}$ | $-0.0767355580$ | $a^{8}_{1}$ | $-0.2253827130$ | $a^{9}_{1}$ | $-0.6095554508$
$a^{7}_{2}$ | $\phantom{-}0.0245616045$ | $a^{8}_{2}$ | $\phantom{-}0.0906682129$ | $a^{9}_{2}$ | $\phantom{-}0.3705808500$
$a^{7}_{3}$ | $-0.0076106356$ | $a^{8}_{3}$ | $-0.0360797100$ | $a^{9}_{3}$ | $-0.2326605853$
$a^{7}_{4}$ | $\phantom{-}0.0023068479$ | $a^{8}_{4}$ | $\phantom{-}0.0141972990$ | $a^{9}_{4}$ | $\phantom{-}0.1434620328$
$a^{7}_{5}$ | $-0.0006627640$ | $a^{8}_{5}$ | $-0.0052036970$ | $a^{9}_{5}$ | $-0.0765436603$
$a^{7}_{6}$ | $\phantom{-}0.0001576564$ | $a^{8}_{6}$ | $\phantom{-}0.0014967921$ | $a^{9}_{6}$ | $\phantom{-}0.0286722918$
$a^{7}_{7}$ | $-0.0000215279$ | $a^{8}_{7}$ | $-0.0002316038$ | $a^{9}_{7}$ | $-0.0052379012$
$s_{7}$ | $\phantom{-}0.1$ | $s_{8}$ | $\phantom{-}0.3$ | $s_{9}$ | $\phantom{-}0.5$
$g_{1}$ | $g_{3}$ | $g_{4}$
---|---|---
$b^{1}_{0}$ | $\phantom{-}0.8752491756$ | $b^{3}_{0}$ | $\phantom{-}0.8242978103$ | $b^{4}_{0}$ | $\phantom{-}0.9435673145$
$b^{1}_{1}$ | $-0.5862941999$ | $b^{3}_{1}$ | $-0.5673416594$ | $b^{4}_{1}$ | $-0.6108767042$
$b^{1}_{2}$ | $\phantom{-}0.3817924531$ | $b^{3}_{2}$ | $\phantom{-}0.3753849098$ | $b^{4}_{2}$ | $\phantom{-}0.3898322718$
$b^{1}_{3}$ | $-0.2509104702$ | $b^{3}_{3}$ | $-0.2488178772$ | $b^{4}_{3}$ | $-0.2534507734$
$b^{1}_{4}$ | $\phantom{-}0.1594109280$ | $b^{3}_{4}$ | $\phantom{-}0.1587444046$ | $b^{4}_{4}$ | $\phantom{-}0.1601943239$
$b^{1}_{5}$ | $-0.0866514938$ | $b^{3}_{5}$ | $-0.0864514894$ | $b^{4}_{5}$ | $-0.0868797888$
$b^{1}_{6}$ | $\phantom{-}0.0328163145$ | $b^{3}_{6}$ | $\phantom{-}0.0327671507$ | $b^{4}_{6}$ | $\phantom{-}0.0328711749$
$b^{1}_{7}$ | $-0.0060331491$ | $b^{3}_{7}$ | $-0.0060262933$ | $b^{4}_{7}$ | $-0.0060406868$
$s_{1}$ | $\phantom{-}0.4$ | $s_{3}$ | $\phantom{-}0.4$ | $s_{4}$ | $\phantom{-}0.5$
$g_{7}$ | $g_{8}$ | $g_{9}$
---|---|---
$b^{7}_{0}$ | $\phantom{-}0.2183469977$ | $b^{8}_{0}$ | $\phantom{-}0.5232652393$ | $b^{9}_{0}$ | $\phantom{-}1.0058222217$
$b^{7}_{1}$ | $-0.0767355702$ | $b^{8}_{1}$ | $-0.2253827769$ | $b^{9}_{1}$ | $-0.6095557922$
$b^{7}_{2}$ | $\phantom{-}0.0245616149$ | $b^{8}_{2}$ | $\phantom{-}0.0906682634$ | $b^{9}_{2}$ | $\phantom{-}0.3705811764$
$b^{7}_{3}$ | $-0.0076106401$ | $b^{8}_{3}$ | $-0.0360797369$ | $b^{9}_{3}$ | $-0.2326608405$
$b^{7}_{4}$ | $\phantom{-}0.0023068496$ | $b^{8}_{4}$ | $\phantom{-}0.0141973115$ | $b^{9}_{4}$ | $\phantom{-}0.1434622111$
$b^{7}_{5}$ | $-0.0006627646$ | $b^{8}_{5}$ | $-0.0052037021$ | $b^{9}_{5}$ | $-0.0765437625$
$b^{7}_{6}$ | $\phantom{-}0.0001576566$ | $b^{8}_{6}$ | $\phantom{-}0.0014967937$ | $b^{9}_{6}$ | $\phantom{-}0.0286723317$
$b^{7}_{7}$ | $-0.0000215279$ | $b^{8}_{7}$ | $-0.0002316041$ | $b^{9}_{7}$ | $-0.0052379086$
$s_{7}$ | $\phantom{-}0.1$ | $s_{8}$ | $\phantom{-}0.3$ | $s_{9}$ | $\phantom{-}0.5$

####  A.1.4 Part 4: $(\mathcal{M}\setminus\mathcal{L})\cap(3.92,4.01)$

It follows from the equalities between columns of the transition matrix $M$
(25), that components of the test function
$\underline{f}=(f_{111},\ldots,f_{333})$ and
$\underline{g}=(g_{111},\ldots,g_{333})$ used in (27) satisfy the following
identities.

$\displaystyle f_{111}$
$\displaystyle=f_{112}=f_{113}=\sum_{k=0}^{7}a_{k}^{111}x^{k}$ $\displaystyle
g_{111}=g_{112}=g_{113}=\sum_{k=0}^{7}b_{k}^{111}x^{k},$ $\displaystyle
f_{211}$ $\displaystyle=f_{2rs}=\sum_{k=0}^{7}a_{k}^{211}x^{k}$ $\displaystyle
g_{211}=g_{2rs}=\sum_{k=0}^{7}b_{k}^{211}x^{k},\ 1\leq r,s\leq 3,$
$\displaystyle f_{121}$
$\displaystyle=f_{122}=f_{123}=\sum_{k=0}^{7}a_{k}^{121}x^{k}$ $\displaystyle
g_{121}=g_{122}=g_{123}=\sum_{k=0}^{7}b_{k}^{121}x^{k},$ $\displaystyle
f_{321}$ $\displaystyle=f_{322}=f_{323}=\sum_{k=0}^{7}a_{k}^{123}x^{k}$
$\displaystyle g_{321}=g_{322}=g_{323}=\sum_{k=0}^{7}b_{k}^{123}x^{k},$
$\displaystyle f_{331}$
$\displaystyle=f_{332}=f_{333}=\sum_{k=0}^{7}a_{k}^{133}x^{k}$ $\displaystyle
g_{331}=g_{332}=g_{333}=\sum_{k=0}^{7}b_{k}^{133}x^{k}.$

Straightforward calculation shows that the functions $f_{j}$ and $g_{j}$ are
monotone decreasing on $[0,1]$ and achieve their minima at $1$. For
convenience, we give a lower bound $s_{qrs}<\min(f_{qrs}(1),g_{qrs}(1))$.

$f_{111}$ | $f_{121}$ | $f_{132}$
---|---|---
$a_{0}^{111}$ | $\phantom{-}0.9884535079$ | $a_{0}^{121}$ | $\phantom{-}0.9249363898$ | $a_{0}^{132}$ | $\phantom{-}0.5745876575$
$a_{1}^{111}$ | $-0.6902235127$ | $a_{1}^{121}$ | $-0.6661893079$ | $a_{1}^{132}$ | $-0.4908751097$
$a_{2}^{111}$ | $\phantom{-}0.4629840472$ | $a_{2}^{121}$ | $\phantom{-}0.4549027221$ | $a_{2}^{132}$ | $\phantom{-}0.3761466270$
$a_{3}^{111}$ | $-0.3124708511$ | $a_{3}^{121}$ | $-0.3098685263$ | $a_{3}^{132}$ | $-0.2757231992$
$a_{4}^{111}$ | $\phantom{-}0.2030567394$ | $a_{4}^{121}$ | $\phantom{-}0.2022427698$ | $a_{4}^{132}$ | $\phantom{-}0.1878939498$
$a_{5}^{111}$ | $-0.1122167788$ | $a_{5}^{121}$ | $-0.1119770597$ | $a_{5}^{132}$ | $-0.1064665719$
$a_{6}^{111}$ | $\phantom{-}0.0429529760$ | $a_{6}^{121}$ | $\phantom{-}0.0428949582$ | $a_{6}^{132}$ | $\phantom{-}0.0412628171$
$a_{7}^{111}$ | $-0.0079480141$ | $a_{7}^{121}$ | $-0.0079400087$ | $a_{7}^{132}$ | $-0.0076832222$
$s_{111}$ | $\phantom{-}0.5$ | $s_{121}$ | $\phantom{-}0.4$ | $s_{132}$ | $\phantom{-}0.2$
$f_{133}$ | $f_{211}$ | $f_{311}$
---|---|---
$a_{0}^{133}$ | $\phantom{-}0.8813943456$ | $a_{0}^{211}$ | $1.0003583962$ | $a_{0}^{311}$ | $\phantom{-}0.5081203394$
$a_{1}^{133}$ | $-0.6491441142$ | $a_{1}^{211}$ | $-0.6654322134$ | $a_{1}^{311}$ | $-0.2326160640$
$a_{2}^{133}$ | $\phantom{-}0.4489736082$ | $a_{2}^{211}$ | $\phantom{-}0.4295333392$ | $a_{2}^{311}$ | $\phantom{-}0.0972736889$
$a_{3}^{133}$ | $-0.3078936043$ | $a_{3}^{211}$ | $-0.2812396746$ | $a_{3}^{311}$ | $-0.0398841238$
$a_{4}^{133}$ | $\phantom{-}0.2016042974$ | $a_{4}^{211}$ | $\phantom{-}0.1785878079$ | $a_{4}^{311}$ | $\phantom{-}0.0160797615$
$a_{5}^{133}$ | $-0.1117833249$ | $a_{5}^{211}$ | $-0.0971302423$ | $a_{5}^{311}$ | $-0.0060041586$
$a_{6}^{133}$ | $\phantom{-}0.0428469743$ | $a_{6}^{211}$ | $\phantom{-}0.0368077301$ | $a_{6}^{311}$ | $\phantom{-}0.0017487411$
$a_{7}^{133}$ | $-0.0079332871$ | $a_{7}^{211}$ | $-0.0067699556$ | $a_{7}^{311}$ | $-0.0002726247$
$s_{133}$ | $\phantom{-}0.4$ | $s_{211}$ | $\phantom{-}0.5$ | $s_{311}$ | $\phantom{-}0.3$
$f_{312}$ | $f_{321}$ | $f_{331}$
---|---|---
$a_{0}^{312}$ | $\phantom{-}0.2013136513$ | $a_{0}^{321}$ | $\phantom{-}0.8072631514$ | $a_{0}^{331}$ | $\phantom{-}1.0061848844$
$a_{1}^{312}$ | $-0.0743470595$ | $a_{1}^{321}$ | $-0.4648188145$ | $a_{1}^{331}$ | $-0.6476491585$
$a_{2}^{312}$ | $\phantom{-}0.0244467077$ | $a_{2}^{321}$ | $\phantom{-}0.2575884636$ | $a_{2}^{331}$ | $\phantom{-}0.4067501349$
$a_{3}^{312}$ | $-0.0077137186$ | $a_{3}^{321}$ | $-0.1452652325$ | $a_{3}^{331}$ | $-0.2603687665$
$a_{4}^{312}$ | $\phantom{-}0.0023694138$ | $a_{4}^{321}$ | $\phantom{-}0.0809083385$ | $a_{4}^{331}$ | $\phantom{-}0.1624071150$
$a_{5}^{312}$ | $-0.0006874054$ | $a_{5}^{321}$ | $-0.0398683005$ | $a_{5}^{331}$ | $-0.0872158384$
$a_{6}^{312}$ | $\phantom{-}0.0001645838$ | $a_{6}^{321}$ | $\phantom{-}0.0141619841$ | $a_{6}^{331}$ | $\phantom{-}0.0327839671$
$a_{7}^{312}$ | $-0.0000225598$ | $a_{7}^{321}$ | $-0.0025036826$ | $a_{7}^{331}$ | $-0.0060002449$
$s_{312}$ | $\phantom{-}0.1$ | $s_{321}$ | $\phantom{-}0.4$ | $s_{331}$ | $\phantom{-}0.5$
$g_{111}$ | $g_{121}$ | $g_{132}$
---|---|---
$b_{0}^{111}$ | $\phantom{-}0.9884535050$ | $b_{0}^{121}$ | $\phantom{-}0.9249364233$ | $b_{0}^{132}$ | $\phantom{-}0.5745877791$
$b_{1}^{111}$ | $-0.6902239809$ | $b_{1}^{121}$ | $-0.6661897749$ | $b_{1}^{132}$ | $-0.4908755105$
$b_{2}^{111}$ | $\phantom{-}0.4629845398$ | $b_{2}^{121}$ | $\phantom{-}0.4549032114$ | $b_{2}^{132}$ | $\phantom{-}0.3761470588$
$b_{3}^{111}$ | $-0.3124712630$ | $b_{3}^{121}$ | $-0.3098689364$ | $b_{3}^{132}$ | $-0.2757235761$
$b_{4}^{111}$ | $\phantom{-}0.2030570414$ | $b_{4}^{121}$ | $\phantom{-}0.2022430710$ | $b_{4}^{132}$ | $\phantom{-}0.1878942344$
$b_{5}^{111}$ | $-0.1122169575$ | $b_{5}^{121}$ | $-0.1119772381$ | $b_{5}^{132}$ | $-0.1064667431$
$b_{6}^{111}$ | $\phantom{-}0.0429530470$ | $b_{6}^{121}$ | $\phantom{-}0.0428950291$ | $b_{6}^{132}$ | $\phantom{-}0.0412628857$
$b_{7}^{111}$ | $-0.0079480275$ | $b_{7}^{121}$ | $-0.0079400220$ | $b_{7}^{132}$ | $-0.0076832353$
$s_{111}$ | $\phantom{-}0.5$ | $s_{121}$ | $\phantom{-}0.4$ | $s_{132}$ | $\phantom{-}0.2$
$g_{133}$ | $g_{211}$ | $g_{311}$
---|---|---
$b_{0}^{133}$ | $\phantom{-}0.8813944143$ | $b_{0}^{211}$ | $1.0003583991$ | $b_{0}^{311}$ | $\phantom{-}0.5081201992$
$b_{1}^{133}$ | $-0.6491445844$ | $b_{1}^{211}$ | $-0.6654326742$ | $b_{1}^{311}$ | $-0.2326161465$
$b_{2}^{133}$ | $\phantom{-}0.4489740965$ | $b_{2}^{211}$ | $\phantom{-}0.4295338070$ | $b_{2}^{311}$ | $\phantom{-}0.0972737581$
$b_{3}^{133}$ | $-0.3078940138$ | $b_{3}^{211}$ | $-0.2812400555$ | $b_{3}^{311}$ | $-0.0398841619$
$b_{4}^{133}$ | $\phantom{-}0.2016045985$ | $b_{4}^{211}$ | $\phantom{-}0.1785880816$ | $b_{4}^{311}$ | $\phantom{-}0.0160797798$
$b_{5}^{133}$ | $-0.1117835035$ | $b_{5}^{211}$ | $-0.0971304019$ | $b_{5}^{311}$ | $-0.0060041662$
$b_{6}^{133}$ | $\phantom{-}0.0428470454$ | $b_{6}^{211}$ | $\phantom{-}0.0368077929$ | $b_{6}^{311}$ | $\phantom{-}0.0017487435$
$b_{7}^{133}$ | $-0.0079333006$ | $b_{7}^{211}$ | $-0.0067699674$ | $b_{7}^{311}$ | $-0.0002726251$
$s_{133}$ | $\phantom{-}0.4$ | $s_{211}$ | $\phantom{-}0.5$ | $s_{311}$ | $\phantom{-}0.3$
$g_{312}$ | $g_{321}$ | $g_{331}$
---|---|---
$b_{0}^{312}$ | $\phantom{-}0.2013135639$ | $b_{0}^{321}$ | $\phantom{-}0.8072631281$ | $b_{0}^{331}$ | $1.0061848887$
$b_{1}^{312}$ | $-0.0743470726$ | $b_{1}^{321}$ | $-0.4648191338$ | $b_{1}^{331}$ | $-0.6476496175$
$b_{2}^{312}$ | $\phantom{-}0.0244467204$ | $b_{2}^{321}$ | $\phantom{-}0.2575887534$ | $b_{2}^{331}$ | $\phantom{-}0.4067505913$
$b_{3}^{312}$ | $-0.0077137243$ | $b_{3}^{321}$ | $-0.1452654416$ | $b_{3}^{331}$ | $-0.2603691317$
$b_{4}^{312}$ | $\phantom{-}0.0023694160$ | $b_{4}^{321}$ | $\phantom{-}0.0809084732$ | $b_{4}^{331}$ | $\phantom{-}0.1624073740$
$b_{5}^{312}$ | $-0.0006874061$ | $b_{5}^{321}$ | $-0.0398683727$ | $b_{5}^{331}$ | $-0.0872159881$
$b_{6}^{312}$ | $\phantom{-}0.0001645840$ | $b_{6}^{321}$ | $\phantom{-}0.0141620110$ | $b_{6}^{331}$ | $\phantom{-}0.0327840257$
$b_{7}^{312}$ | $-0.0000225598$ | $b_{7}^{321}$ | $-0.0025036874$ | $b_{7}^{331}$ | $-0.0060002559$
$s_{312}$ | $\phantom{-}0.1$ | $s_{321}$ | $\phantom{-}0.4$ | $s_{331}$ | $\phantom{-}0.5$

####  A.1.5 Part 5:
$(\mathcal{M}\setminus\mathcal{L})\cap(\sqrt{20},\sqrt{21})$

We present coefficients of the polynomial components of test functions
$\underline{f}=(f_{1},f_{2},f_{3},f_{4})$ and
$\underline{g}=(g_{1},g_{2},g_{3},g_{4})$ used in (30). It follows from the
equality between the first and second columns of the matrix $M$ that
$f_{1}=f_{2}$ and $g_{1}=g_{2}$.

$f_{j}=\sum_{k=0}^{9}a_{k}^{j}x^{k}\quad
g_{j}=\sum_{k=0}^{9}b_{k}^{j}x^{k},\quad j=1,2,3,4.$

Straightforward calculation shows that the functions $f_{j}$ and $g_{j}$ are
monotone decreasing on $[0,1]$ and achieve their minima at $1$. For
convenience, we give a lower bound $s_{j}<\min(f_{j}(1),g_{j}(1))$.

$f_{1}$ | $f_{3}$ | $f_{4}$
---|---|---
$a^{1}_{0}$ | $\phantom{-}0.9799928531$ | $a^{3}_{0}$ | $\phantom{-}1.0045893915$ | $a^{4}_{0}$ | $\phantom{-}0.1934516264$
$a^{1}_{1}$ | $-0.7406258897$ | $a^{3}_{1}$ | $-0.7487821988$ | $a^{4}_{1}$ | $-0.0765314787$
$a^{1}_{2}$ | $\phantom{-}0.5273926877$ | $a^{3}_{2}$ | $\phantom{-}0.5296982401$ | $a^{4}_{2}$ | $\phantom{-}0.0259667329$
$a^{1}_{3}$ | $-0.3800229054$ | $a^{3}_{3}$ | $-0.3806370202$ | $a^{4}_{3}$ | $-0.0083489461$
$a^{1}_{4}$ | $\phantom{-}0.2763248479$ | $a^{3}_{4}$ | $\phantom{-}0.2764834085$ | $a^{4}_{4}$ | $\phantom{-}0.0026152698$
$a^{1}_{5}$ | $-0.1955487122$ | $a^{3}_{5}$ | $-0.1955888334$ | $a^{4}_{5}$ | $-0.0008056462$
$a^{1}_{6}$ | $\phantom{-}0.1241717000$ | $a^{3}_{6}$ | $\phantom{-}0.1241816277$ | $a^{4}_{6}$ | $\phantom{-}0.0002419581$
$a^{1}_{7}$ | $-0.0622409671$ | $a^{3}_{7}$ | $-0.0622432782$ | $a^{4}_{7}$ | $-0.0000670179$
$a^{1}_{8}$ | $\phantom{-}0.0206092394$ | $a^{3}_{8}$ | $\phantom{-}0.0206096817$ | $a^{4}_{8}$ | $\phantom{-}0.0000146636$
$a^{1}_{9}$ | $-0.0032442178$ | $a^{3}_{9}$ | $-0.0032442666$ | $a^{4}_{9}$ | $-0.0000017681$
$s_{1}$ | $\phantom{-}0.4$ | $s_{3}$ | $\phantom{-}0.4$ | $s_{4}$ | $\phantom{-}0.05$
$g_{1}$ | $g_{3}$ | $g_{4}$
---|---|---
$b^{1}_{0}$ | $\phantom{-}0.9799928382$ | $b^{3}_{0}$ | $\phantom{-}1.0045893900$ | $b^{4}_{0}$ | $\phantom{-}0.1934516765$
$b^{1}_{1}$ | $-0.7406256446$ | $b^{3}_{1}$ | $-0.7487819559$ | $b^{4}_{1}$ | $-0.0765314768$
$b^{1}_{2}$ | $\phantom{-}0.5273924149$ | $b^{3}_{2}$ | $\phantom{-}0.5296979674$ | $b^{4}_{2}$ | $\phantom{-}0.0259667280$
$b^{1}_{3}$ | $-0.3800226607$ | $b^{3}_{3}$ | $-0.3806367755$ | $b^{4}_{3}$ | $-0.0083489435$
$b^{1}_{4}$ | $\phantom{-}0.2763246450$ | $b^{3}_{4}$ | $\phantom{-}0.2764832054$ | $b^{4}_{4}$ | $\phantom{-}0.0026152687$
$b^{1}_{5}$ | $-0.1955485561$ | $b^{3}_{5}$ | $-0.1955886769$ | $b^{4}_{5}$ | $-0.0008056458$
$b^{1}_{6}$ | $\phantom{-}0.1241715958$ | $b^{3}_{6}$ | $\phantom{-}0.1241815227$ | $b^{4}_{6}$ | $\phantom{-}0.0002419578$
$b^{1}_{7}$ | $-0.0622409134$ | $b^{3}_{7}$ | $-0.0622432239$ | $b^{4}_{7}$ | $-0.0000670177$
$b^{1}_{8}$ | $\phantom{-}0.0206092213$ | $b^{3}_{8}$ | $\phantom{-}0.0206096634$ | $b^{4}_{8}$ | $\phantom{-}0.0000146636$
$b^{1}_{9}$ | $-0.0032442150$ | $b^{3}_{9}$ | $-0.0032442637$ | $b^{4}_{9}$ | $-0.0000017681$
$s_{1}$ | $\phantom{-}0.4$ | $s_{3}$ | $\phantom{-}0.4$ | $s_{4}$ | $\phantom{-}0.05$

###  A.2 Zaremba theory

####  A.2.1 $\dim_{H}(E_{5})$

We present coefficients of the polynomial test functions $f$ and $g$ used in
(36).

$f=\sum_{k=0}^{15}a_{k}x^{k},\quad g=\sum_{k=0}^{15}b_{k}x^{k}.$

Similarly to the previous examples, the functions $f$ and $g$ are monotone and
they can be bounded from below by their value at $1$. In particular, we have
$f(1),g(1)\geq 0.4$.

$f$ | $g$
---|---
$a_{0}$ | $\phantom{-}1.002075775192587$ | $b_{0}$ | $\phantom{-}1.002075775192580$
$a_{1}$ | $-0.863832791554195$ | $b_{1}$ | $-0.863832791551160$
$a_{2}$ | $\phantom{-}0.694904605500679$ | $b_{2}$ | $\phantom{-}0.694904605500778$
$a_{3}$ | $-0.563982609501401$ | $b_{3}$ | $-0.563982609938672$
$a_{4}$ | $\phantom{-}0.462936795376894$ | $b_{4}$ | $\phantom{-}0.462936803438964$
$a_{5}$ | $-0.382787786278096$ | $b_{5}$ | $-0.382787860060200$
$a_{6}$ | $\phantom{-}0.317798375235142$ | $b_{6}$ | $\phantom{-}0.317798788806115$
$a_{7}$ | $-0.263717774563333$ | $b_{7}$ | $-0.263719322983292$
$a_{8}$ | $\phantom{-}0.216187532791877$ | $b_{8}$ | $\phantom{-}0.216191581202906$
$a_{9}$ | $-0.170354610076857$ | $b_{9}$ | $-0.170362167218400$
$a_{10}$ | $\phantom{-}0.123046705346496$ | $b_{10}$ | $\phantom{-}0.123056842712685$
$a_{11}$ | $-0.076412879097916$ | $b_{11}$ | $-0.076422574631579$
$a_{12}$ | $\phantom{-}0.037844499485800$ | $b_{12}$ | $\phantom{-}0.037850946915568$
$a_{13}$ | $-0.013650349050295$ | $b_{13}$ | $-0.013653178641107$
$a_{14}$ | $\phantom{-}0.003133011516183$ | $b_{14}$ | $\phantom{-}0.003133747261017$
$a_{15}$ | $-0.000339760445059$ | $b_{15}$ | $-0.000339846126736$

####  A.2.2 $\dim_{H}(E_{4})$

Coefficients of the polynomial test functions $f$ and $g$ used in (37).

$f=\sum_{k=0}^{15}a_{k}x^{k},\quad g=\sum_{k=0}^{15}b_{k}x^{k}.$

Similar to the previous examples, the functions $f$ and $g$ are monotone and
they can be bounded from below by their value at $1$. In particular, we have
$f(1),g(1)\geq 0.4$.

$f$ | $g$
---|---
$a_{0}$ | $\phantom{-}1.001981557057916$ | $b_{0}$ | $\phantom{-}1.001981557057906$
$a_{1}$ | $-0.824549641777407$ | $b_{1}$ | $-0.824549641773357$
$a_{2}$ | $\phantom{-}0.632377740413372$ | $b_{2}$ | $\phantom{-}0.632377740394589$
$a_{3}$ | $-0.489751892709023$ | $b_{3}$ | $-0.489751892404770$
$a_{4}$ | $\phantom{-}0.384644919648888$ | $b_{4}$ | $\phantom{-}0.384644915987248$
$a_{5}$ | $-0.305093638471734$ | $b_{5}$ | $-0.305093610668140$
$a_{6}$ | $\phantom{-}0.243481281464752$ | $b_{6}$ | $\phantom{-}0.243481142895873$
$a_{7}$ | $-0.194635007255869$ | $b_{7}$ | $-0.194634539676458$
$a_{8}$ | $\phantom{-}0.154196605135780$ | $b_{8}$ | $\phantom{-}0.154195521320957$
$a_{9}$ | $-0.118018014953805$ | $b_{9}$ | $-0.118016296613633$
$a_{10}$ | $\phantom{-}0.083325052299187$ | $b_{10}$ | $\phantom{-}0.083323249134991$
$a_{11}$ | $-0.050893608677143$ | $b_{11}$ | $-0.050892477178423$
$a_{12}$ | $\phantom{-}0.024910421954700$ | $b_{12}$ | $\phantom{-}0.024910155247198$
$a_{13}$ | $-0.008908424701076$ | $b_{13}$ | $-0.008908569288906$
$a_{14}$ | $\phantom{-}0.002031148876994$ | $b_{14}$ | $\phantom{-}0.002031269948929$
$a_{15}$ | $-0.000219053588808$ | $b_{15}$ | $-0.000219079665840$

####  A.2.3 $\dim_{H}(E_{1235})$

Coefficients of the polynomial test functions $f$ and $g$ corresponding to
$t_{1}=0.7709149399\,36$ and $t_{0}=t_{1}+3\cdot 10^{-12}$ respectively.

$f=\sum_{k=0}^{15}a_{k}x^{k},\quad g=\sum_{k=0}^{15}b_{k}x^{k}.$

Similar to the previous examples, the functions $f$ and $g$ are monotone and
they can be bounded from below by their value at $1$. In particular, we have
$f(1),g(1)\geq 0.4$.

$f$ | $g$
---|---
$a_{0}$ | $\phantom{-}1.001943825796940$ | $b_{0}$ | $\phantom{-}1.001943825796930$
$a_{1}$ | $-0.808837588421526$ | $b_{1}$ | $-0.808837588417559$
$a_{2}$ | $\phantom{-}0.615494218966991$ | $b_{2}$ | $\phantom{-}0.615494218954915$
$a_{3}$ | $-0.474372511899657$ | $b_{3}$ | $-0.474372511806688$
$a_{4}$ | $\phantom{-}0.371576736077143$ | $b_{4}$ | $\phantom{-}0.371576735828591$
$a_{5}$ | $-0.294687522275668$ | $b_{5}$ | $-0.294687526007311$
$a_{6}$ | $\phantom{-}0.235767134415525$ | $b_{6}$ | $\phantom{-}0.235767178162078$
$a_{7}$ | $-0.189403864881314$ | $b_{7}$ | $-0.189404097267897$
$a_{8}$ | $\phantom{-}0.151071322536495$ | $b_{8}$ | $\phantom{-}0.151072090158663$
$a_{9}$ | $-0.116493763140853$ | $b_{9}$ | $-0.116495485397309$
$a_{10}$ | $\phantom{-}0.082822264062997$ | $b_{10}$ | $\phantom{-}0.082824975856055$
$a_{11}$ | $-0.050871691397334$ | $b_{11}$ | $-0.050874703061709$
$a_{12}$ | $\phantom{-}0.025002624053740$ | $b_{12}$ | $\phantom{-}0.025004941184306$
$a_{13}$ | $-0.008966855413747$ | $b_{13}$ | $-0.008968032838312$
$a_{14}$ | $\phantom{-}0.002048316877335$ | $b_{14}$ | $\phantom{-}0.002048672642558$
$a_{15}$ | $-0.000221169553698$ | $b_{15}$ | $-0.000221217982471$

###  A.3 Fuchsian Schottky groups

Coefficients of the polynomial test functions
$\underline{f}=(f_{0},f_{1},f_{2})$ and $\underline{g}=(g_{0},g_{1},g_{2})$,
where each $f_{j}$ and $g_{j}$ for $j=0,1,2$ is defined on the interval
$X_{j}$, respectively.

$f_{j}=\sum_{k=0}^{24}a_{k}^{j}x^{k},\quad
g_{j}=\sum_{k=0}^{24}b_{k}^{j}x^{k},\quad j=0,1,2.$

The functions $f_{0}$ and $g_{0}$ are even and therefore their odd
coefficients vanish: $a^{0}_{2k+1}=b^{0}_{2k+1}\equiv 0$. Moreover,
$f_{1}(x)=f_{2}(-x)$ and $g_{1}(x)=g_{2}(-x)$ therefore their even
coefficients agree and their odd coefficients have the opposite signs:
$a^{1}_{2k}=a^{2}_{2k}$, $a^{1}_{2k+1}=-a^{2}_{2k+1}$;
$b^{1}_{2k}=b^{2}_{2k}$, $b^{1}_{2k+1}=-b^{2}_{2k+1}$.

$f_{1}$
---
$a^{1}_{0}$ | $2.8770704827462130787849251492030$
$a^{1}_{1}$ | $-11.4401515732168029128293331165380$
$a^{1}_{2}$ | $41.2783414403998127337761764215520$
$a^{1}_{3}$ | $-111.0688407519945991015658074607160$
$a^{1}_{4}$ | $229.2359826830719504436183231429020$
$a^{1}_{5}$ | $-374.1839317918421651508580498351190$
$a^{1}_{6}$ | $494.2486182891231866860550011193190$
$a^{1}_{7}$ | $-537.1550718818002043450127874148360$
$a^{1}_{8}$ | $486.1852872370976132985936371604760$
$a^{1}_{9}$ | $-369.6791357119376158018979254258860$
$a^{1}_{10}$ | $237.5656093180881359651809836821330$
$a^{1}_{11}$ | $-129.5193450984147750120850142699940$
$a^{1}_{12}$ | $60.0196876244694040557343257845630$
$a^{1}_{13}$ | $-23.6421320534647605005954912805860$
$a^{1}_{14}$ | $7.9021374699139922100387246571970$
$a^{1}_{15}$ | $-2.2327574635963721814894503262740$
$a^{1}_{16}$ | $0.5301045742621054501101376372390$
$a^{1}_{17}$ | $-0.1048257433528365630840057396770$
$a^{1}_{18}$ | $0.0170498220734007394058868515300$
$a^{1}_{19}$ | $-0.0022409149968744747201392201870$
$a^{1}_{20}$ | $0.0002320048619867762156560576330$
$a^{1}_{21}$ | $-0.0000182076188328122444526479340$
$a^{1}_{22}$ | $1.017711979517475899435180\cdot 10^{-6}\phantom{18}$
$a^{1}_{23}$ | $-3.60864141942206682494120\cdot 10^{-8}\phantom{188}$
$a^{1}_{24}$ | $6.09941161252684275320\cdot 10^{-10}\phantom{88888}$
$f_{0}$
$a^{0}_{0}$ | $0.988509120501286981726492299097$
$a^{0}_{2}$ | $0.157542850847455232142931716289$
$a^{0}_{4}$ | $0.042904006074342082007020639894$
$a^{0}_{6}$ | $0.013311224077669383683707536427$
$a^{0}_{8}$ | $0.004429850096065885316293787561$
$a^{0}_{10}$ | $0.001543440394063281474032350116$
$a^{0}_{12}$ | $0.000555351173373256765356036270$
$a^{0}_{14}$ | $0.000204517408718030083947390234$
$a^{0}_{16}$ | $0.000076609680108249015238001359$
$a^{0}_{18}$ | $0.000029061155169646923244681761$
$a^{0}_{20}$ | $0.000011137276040685233840494967$
$a^{0}_{22}$ | $4.2275802352717104321010360\cdot 10^{-6}$
$a^{0}_{24}$ | $1.9876322730511567993854040\cdot 10^{-6}$

$g_{1}$
---
$b^{1}_{0}$ | $2.877070483741637235339396793709$
$b^{1}_{1}$ | $-11.440151580102508283298777329005$
$b^{1}_{2}$ | $41.278341467154591279141888564443$
$b^{1}_{3}$ | $-111.068840826093040583068331287781$
$b^{1}_{4}$ | $229.235982838419674330781933540686$
$b^{1}_{5}$ | $-374.183932047938578510658766660908$
$b^{1}_{6}$ | $494.248618629707611522216515713345$
$b^{1}_{7}$ | $-537.155072253799253891555878102640$
$b^{1}_{8}$ | $486.185287575077537070728753265866$
$b^{1}_{9}$ | $-369.679135969693151986947450814820$
$b^{1}_{10}$ | $237.565609484126480000044027316791$
$b^{1}_{11}$ | $-129.519345189116330520934728232633$
$b^{1}_{12}$ | $60.019687666569911991602610502470$
$b^{1}_{13}$ | $-23.642132070071466367919223736091$
$b^{1}_{14}$ | $7.902137475471208082330326553248$
$b^{1}_{15}$ | $-2.232757465168180629595168192690$
$b^{1}_{16}$ | $0.530104574635619914117715916573$
$b^{1}_{17}$ | $-0.104825743426755206788900861842$
$b^{1}_{18}$ | $0.017049822085431863890249313609$
$b^{1}_{19}$ | $-0.002240914998456736922493194695$
$b^{1}_{20}$ | $0.000232004862150679716335333676$
$b^{1}_{21}$ | $-0.000018207618845681630026751529$
$b^{1}_{22}$ | $1.0177119802371267732818260\cdot 10^{-6}$
$b^{1}_{23}$ | $-3.60864142197485386059250\cdot 10^{-8}$
$b^{1}_{24}$ | $6.099411616843192081320\cdot 10^{-10}$
$g_{0}$
$b^{0}_{0}$ | $0.988509120496506332431826934199$
$b^{0}_{2}$ | $0.157542850912623683440552637873$
$b^{0}_{4}$ | $0.042904006097255254396958615314$
$b^{0}_{6}$ | $0.013311224085738465758389603856$
$b^{0}_{8}$ | $0.004429850098979983540304963280$
$b^{0}_{10}$ | $0.001543440395140704805733043742$
$b^{0}_{12}$ | $0.000555351173779220956618567711$
$b^{0}_{14}$ | $0.000204517408873232993380371066$
$b^{0}_{16}$ | $0.000076609680168235804587352199$
$b^{0}_{18}$ | $0.000029061155193021143267002387$
$b^{0}_{20}$ | $0.000011137276049855568024571829$
$b^{0}_{22}$ | $4.2275802388232323523285170\cdot 10^{-6}$
$b^{0}_{24}$ | $1.9876322747582711067418060\cdot 10^{-6}$

###  A.4 Non-linear example

Coefficients of the polynomial test functions $f$ and $g$ corresponding to
$t_{0}=0.4934480908\,02$ and $t_{1}=t_{0}+10^{-12}$ respectively.

$f=\sum_{k=0}^{7}a_{2k}x^{2k},\quad g=\sum_{k=0}^{7}b_{2k}x^{2k}.$ $f$ | $g$
---|---
$a_{0}$ | $\phantom{-}0.260509445190371$ | $b_{0}$ | $\phantom{-}0.260509445190371$
$a_{2}$ | $-4.88560887219182\cdot 10^{-3}$ | $b_{2}$ | $-4.88560887219092\cdot 10^{-3}$
$a_{4}$ | $\phantom{-}3.65942002704976\cdot 10^{-4}$ | $b_{4}$ | $\phantom{-}3.65942002704768\cdot 10^{-4}$
$a_{6}$ | $-3.48521776330216\cdot 10^{-5}$ | $b_{6}$ | $-3.48521776329928\cdot 10^{-5}$
$a_{8}$ | $\phantom{-}3.61101766657243\cdot 10^{-6}$ | $b_{8}$ | $\phantom{-}3.61101766656879\cdot 10^{-6}$
$a_{10}$ | $-3.88502731592718\cdot 10^{-7}$ | $b_{10}$ | $-3.88502731592271\cdot 10^{-7}$
$a_{12}$ | $\phantom{-}4.09696977259504\cdot 10^{-8}$ | $b_{12}$ | $\phantom{-}4.09696977258989\cdot 10^{-8}$
$a_{14}$ | $-3.23465524617897\cdot 10^{-9}$ | $b_{14}$ | $-3.23465524617468\cdot 10^{-9}$

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P. Vytnova, Department of Mathematics, Warwick University, Coventry, CV4 7AL,
UK E-mail address<EMAIL_ADDRESS>
M. Pollicott, Department of Mathematics, Warwick University, Coventry, CV4
7AL, UK. E-mail address<EMAIL_ADDRESS>