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Non-ergodicity of the geodesic flow on Cantor tree surface

A Riemann surface equipped with its conformal hyperbolic metric is parabolic if and only if the geodesic flow on its unit tangent bundle is ergodic. Let X be a Cantor tree or a blooming Cantor tree Riemann surface. Fix a geodesic pants decomposition of X and call the boundary geodesics in the decomposition cuffs. Basmajian, Hakobyan, and Šarić proved that if the lengths of cuffs are rapidly converging to zero, then X is parabolic. More recently, Šarić proved a slightly slower convergence of lengths of cuffs to zero implies X is not parabolic. In the paper, we interpolate between the two rates of convergence of the cuffs to zero and find that these surfaces are not parabolic, thus completing the picture.

## Figure 1. The Cantor tree surface with a geodesic pants decomposition.

A Cantor tree Riemann surface X_{C} is conformal to the complement of a Cantor set in the Riemann sphere. Equivalently, X_{C} is constructed by isometrically gluing countably many geodesic pairs of pants along their boundary geodesics (called cuffs) to form the ``shape'' of the dyadic tree (see Figure 1). In addition to the lengths of its cuffs, the Cantor tree Riemann surface X_{C} is determined by the twists along the cuffs.

The cuffs of X_{C} are grouped in the levels based on the level in the dyadic tree. At level zero, we have a single cuff, which is at the top of X_{C} in Figure 1. At level one, we have four cuffs, and at level n ≥ 1, we have 2^{n+1} cuffs. Denote by {α_{n}^{j}}_{j=1}^{2n+1} the level n cuffs from left to right in Figure 1.

McMullen [McMullen] proved if there is a C > 0 such that 1/C ≤ l(α_{n}^{j}) ≤ C then X is not parabolic. This is a consequence of the fact that the Brownian motion has many directions to escape to infinity when the ideal boundary is large, in our case the Cantor set, and the cuffs (the openings) are not short.

In the case when the cuffs are short Basmajian, Hakobyan and Šarić [BHS] proved X_{C} is not parabolic if there is C > 0 such that l(α_{n}^{j}) ≤ C |
n 2 ^{n} |
, where l(.) is the hyperbolic length in X. |

More recently, Šarić[S, Theorem 8.3] proved that if

l( α_{n}^{j})= |
n ^{r}2 ^{n} |

for r > 2, and for all n ≥ 1 and j=1,2, ... , 2

The remaining case to consider is whether X_{C} is parabolic or not for $1 < r ≤ 2. We show the following.

Theorem 1.1. Let X_{C} be the Cantor tree surface as depicted in Figure 1 and l(α_{n}^{j}) for 1 ≤ j ≤ 2^{n+1} the cuffs at the level n. The cuff lengths are decreasing along each end. Then X_{C} is not parabolic if there is an r > 1 such that |

C_{1} | n ^{r}2 ^{n} | ≤ l(α_{n}^{j}) ≤ | C _{2}n ^{2} |

## Figure 2. The blooming Cantor tree surface with a geodesic pants decomposition.

Even for r > 2, the scope of our theorem is slightly more general than [S, Theorem 8.3] because we allow the lengths of the cuffs to vary with the given lower bound. We also extend our result to surfaces with infinite genus and a Cantor set of ends, called the blooming Cantor tree surfaces X̃_{C} (see Figure 2). To construct the blooming Cantor tree from the Cantor tree, attach a geodesic surface of genus at most C and two boundaries, or do not, to each level n boundary α_{n}^{j} (see that α_{0}^{1}=α_{0}^{2}=α_{0}) and redefine α_{n}^{j} to be the boundary of the attached surface further away from α_{0} for n ≥ 0 and for 1 ≤ j ≤ 2^{n+1} and for a universal constant C>0. We can add 2^{n+1} surfaces at the level n of genus at most C for n ≥ 0. Assume the lengths of cuffs along each attached geodesic surface are decreasing.

C_{1} | n ^{r}2 ^{n} | ≤ l(α_{n}^{j}) ≤ | C _{2}n ^{2} |

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