## Journals

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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

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.

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}.## Year of publication

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