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### Open Access Journals

JMD

In this paper we discuss a coding and the associated symbolic dynamics
for the geodesic flow on Hecke triangle surfaces. We construct an
explicit cross-section for which the first-return map factors through
a simple (explicit) map given in terms of the generating map of a
particular continued-fraction expansion closely related to the Hecke
triangle groups. We also obtain explicit expressions for the associated
first return times.

DCDS

Let $T_{f}$ :

*S*^{1}→*S*^{1}be a circle homeomorphism with two break points a_{b}, c_{b}that means the derivative $Df$ of its lift $f\ :\ \mathbb{R}\rightarrow\mathbb{R}$ has discontinuities at the points ã_{b}, ĉ_{b}, which are the representative points of a_{b}, c_{b}in the interval $[0,1)$, and irrational rotation number ρ_{f}. Suppose that $Df$ is absolutely continuous on every connected interval of the set [0,1]\{ã_{b}, ĉ_{b}}, that DlogDf ∈ L^{1}([0,1]) and the product of the jump ratios of $ Df $ at the break points is nontrivial, i.e. $\frac{Df_{-}(\tilde{a}_{b})}{Df_{+}(\tilde{a}_{b})}\frac{Df_{-}(\tilde{c}_{b})}{Df_{+}(\tilde{c}_{b})} \ne1$. We prove, that the unique T_{f}- invariant probability measure $\mu_{f}$ is then singular with respect to Lebesgue measure on S^{1}.
DCDS

In this paper we extend the transfer operator approach to Selberg's zeta function for cofinite
Fuchsian groups to the Hecke triangle groups $G_q,\, q=3,4,\ldots$, which are non-arithmetic for
$q\not= 3,4,6$. For this we make use of a Poincar\'e map for the geodesic flow on the corresponding
Hecke surfaces, which has been constructed in [13], and which is closely related to the natural extension of the generating map for the
so-called Hurwitz-Nakada continued fractions.
We also derive functional equations for the eigenfunctions of the transfer operator which for
eigenvalues $\rho =1$ are expected to be closely related to the period functions of Lewis and Zagier
for these Hecke triangle groups.

DCDS

Let $f$ be an orientation preserving circle homeomorphism with a
single break point $x_b,$ i.e. with a jump in the first derivative
$f'$ at the point $x_b,$ and with irrational rotation number
$\rho=\rho_{f}.$ Suppose that $f$ satisfies the Katznelson and
Ornstein smoothness conditions, i.e. $f'$ is absolutely continuous
on $[x_b,x_b+1]$ and $f''(x)\in \mathbb{L}^{p}([0,1), d\ell)$ for
some $p>1$, where $\ell$ is Lebesque measure. We prove, that the
renormalizations of $f$ are approximated by linear-fractional
functions in $\mathbb{C}^{1+L^{1}}$, that means, $f$ is approximated
in $C^{1}-$ norm and $f''$ is appoximated in $L^{1}-$ norm.
Also it is shown, that
renormalizations of circle diffeomorphisms with irrational rotation
number satisfying the Katznelson and Ornstein smoothness conditions
are close to linear functions in $\mathbb{C}^{1+L^{1}}$- norm.

keywords:
break point
,
renormalizations
,
Circle homeomorphism
,
rotation number
,
fractional linear maps.

## Year of publication

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