Russian Mathematical Surveys, 2023, Volume 78, Issue 3, Pages 566–568 DOI: https://doi.org/10.4213/rm10099e (Mi rm10099)

DOI: https://doi.org/10.4213/rm10099e

Funding agency	Grant number
European Commission, project INFINITE-CELL	647133 ICHAOS
Russian Foundation for Basic Research	18-51-15010 18-31-20031
The work of A. I. Bufetov received funding from the European Research Council under grant agreement no. 647133 (ICHAOS). The work of A. V. Klimenko was partially supported by the Russian Foundation for Basic Research and the French National Centre for Scientific Research (CNRS) (joint grant no. 18-51-15010) and by the Russian Foundation for Basic Research (grant no. 18-31-20031.

Russian version:
Uspekhi Matematicheskikh Nauk, 2023, Volume 78, Issue 3(471), Pages 179–180
DOI: https://doi.org/10.4213/rm10099

Bibliographic databases:

The main result of this note, Theorem 1, establishes the pointwise convergence of spherical averages for actions of Fuchsian groups.

Let $G$ be a finitely generated Fuchsian group. Let $\mathcal R$ be its fundamental domain, possibly with vertices or edges on the boundary of the hyperbolic disc $ \mathbb D$, and let ${\mathbf{T}_{\mathcal{R}}}=\{g{\mathcal R}\colon g\in G\}$ be the corresponding tessellation of $\mathbb D$. We say that ${\mathcal R}$ has even corners if the geodesic extension of every side of ${\mathcal R}$ if entirely contained in the union of the boundaries of all domains $g{\mathcal R}\in{\mathbf{T}_{\mathcal{R}}}$.

Let $G_0$ be the symmetric set of generators mapping $\mathcal R$ to adjacent domains in ${\mathbf{T}_{\mathcal{R}}}$. For $g\in G$ denote by $|g|$ the length of the shortest word in $G_0$ representing $g$. Let $S(n)$ be the sphere of radius $n$ in $G$: $S(n)=\{g\in G\colon |g|=n\}$.

Suppose that $G$ acts on a probability space $(X, \mu)$ by measure-preserving transformations $T_g$, $g\in G$. Given a function $f\in L^1(X,\mu)$, consider the spherical averages

Set $L\log L(X,\mu)=\biggl\{f \in L^1\colon \displaystyle\int |f| \log ^+|f|\,d\mu < \infty\biggr\}$.

Our proof extends the argument from [3], where the convergence of the spherical averages was established for actions of free groups. The main step in the proof of Theorem 1 is the construction of a new Markov coding for a Fuchsian group satisfying Assumption 1.

The first results on the convergence of spherical averages for Gromov hyperbolic groups, which were obtained under strong exponential mixing assumptions on the action, were due to Fujiwara and Nevo [5]. A convenient method proposed by Grigorchuk [6], Thouvenot (oral communication), and in [2] for proving ergodic theorems for actions of free semigroups and groups is to associate with the group action a Markov operator $P$ on a suitable function space. The convergence of spherical averages is thus related to the convergence of the powers $P^n f$ of this Markov operator. The proof of the convergence for free groups in [3] used Rota’s ‘Alternierende Verfahren’ [7], that is, the convergence of $(P^*)^n P^n f$. To deduce the convergence of $P^{2n}f$ one needs a relation between $P$ and $P^*$: see [3], Proposition 3. The source for this relation is the following symmetry condition for the underlying Markov coding for $G$.

• 1. Let $\Xi$ be the set of states of the coding. Then there exists a time-reversing involution $\iota\colon\Xi\to\Xi$, that is, the transitions $\iota(j)\to\iota(k)$ and $k\to j$ are admissible simultaneously.
• 2. There exist two maps $\gamma,\omega\colon\Xi\to G_0$ such that
  $$ \begin{equation*} \omega(\iota(k))=\omega(k)^{-1};\qquad \gamma(k)=\omega(j)^{-1}\gamma(\iota(j))^{-1} \omega(k) \quad \text{if } k\to j \text{ is admissible}. \end{equation*} \notag $$
• 3. There exist subsets $\Xi_S$ and $\Xi_F$ of $ \Xi$ such that the set of all admissible sequences $j_0\to\cdots\to j_{n-1}$ such that $j_0\in \Xi_S$ and $j_{n-1}\in \Xi_F$ is bijectively mapped onto the sphere $S(n)\subset G$ by the map
  $$ \begin{equation*} (j_0\to\cdots\to j_{n-1})\mapsto\omega(j_{n-1})\gamma(j_{n-2})\cdots \gamma(j_0). \end{equation*} \notag $$

For the general Fuchsian group the earlier coding by Bowen and Series [1], [4] does not satisfy this symmetry condition. Recently Wroten [8] introduced a new approach, which is based on the simultaneous consideration of all shortest words representing a given element $g$. The main step in our proof of Theorem 1 is to show that for Fuchsian groups collections of all shortest paths are generated by a symmetric Markov coding such that the generating Markov chain is irreducible and has a trivial symmetric $\sigma$-algebra.

Bibliography
1.	R. Bowen and C. Series, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 153–170
2.	A. I. Bufetov, Funct. Anal. Appl., 34:4 (2000), 239–251
3.	A. I. Bufetov, Ann. of Math. (2), 155:3 (2002), 929–944
4.	A. I. Bufetov and C. Series, Math. Proc. Cambridge Philos. Soc., 151:1 (2011), 145–159
5.	K. Fujiwara and A. Nevo, Ergodic Theory Dynam. Systems, 18:4 (1998), 843–858
6.	R. I. Grigorchuk, Proc. Steklov Inst. Math., 231 (2000), 113–127
7.	G.-C. Rota, Bull. Amer. Math. Soc., 68 (1962), 95–102
8.	M. Wroten, The eventual Gaussian distribution of self-intersection numbers on closed surfaces, Thesis (Ph.D.), State Univ. of New York, Stony Brook, 2013, 41 pp.  ; 2014, 43 pp., arXiv: 1405.7951

Citation in format AMSBIB

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\by A.~I.~Bufetov, A.~V.~Klimenko, C.~Series
\paper An ergodic theorem for actions of Fuchsian groups
\jour Russian Math. Surveys
\yr 2023
\vol 78
\issue 3
\pages 566--568
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