Symbolic dynamics for the geodesic flow on Hecke surfaces
Dieter Mayer Fredrik Strömberg
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.
keywords: symbolic dynamics Hecke triangle groups continued fractions geodesic flow
The transfer operator for the Hecke triangle groups
Dieter Mayer Tobias Mühlenbruch Fredrik Strömberg
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.
keywords: $\lambda_q$-continued fractions Ruelle and Selberg zeta function. Hecke triangle groups transfer operator
Renormalizations of circle hoemomorphisms with a single break point
Abdumajid Begmatov Akhtam Dzhalilov Dieter Mayer
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.
Singular measures of piecewise smooth circle homeomorphisms with two break points
Akhtam Dzhalilov Isabelle Liousse Dieter Mayer
Let $T_{f}$ : S1S1 be a circle homeomorphism with two break points ab, cb that means the derivative $Df$ of its lift $f\ :\ \mathbb{R}\rightarrow\mathbb{R}$ has discontinuities at the points ã b, ĉb, which are the representative points of ab, cb in the interval $[0,1)$, and irrational rotation number ρf. Suppose that $Df$ is absolutely continuous on every connected interval of the set [0,1]\{ãb, ĉb}, that DlogDf ∈ L1([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 Tf - invariant probability measure $\mu_{f}$ is then singular with respect to Lebesgue measure on S1.
keywords: Circle homeomorphism rotation number invariant measures. break points

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