Symbolic dynamics for the geodesic flow on Hecke surfaces
Dieter Mayer Fredrik Strömberg
Journal of Modern Dynamics 2008, 2(4): 581-627 doi: 10.3934/jmd.2008.2.581
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
Singular measures of piecewise smooth circle homeomorphisms with two break points
Akhtam Dzhalilov Isabelle Liousse Dieter Mayer
Discrete & Continuous Dynamical Systems - A 2009, 24(2): 381-403 doi: 10.3934/dcds.2009.24.381
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
The transfer operator for the Hecke triangle groups
Dieter Mayer Tobias Mühlenbruch Fredrik Strömberg
Discrete & Continuous Dynamical Systems - A 2012, 32(7): 2453-2484 doi: 10.3934/dcds.2012.32.2453
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
Discrete & Continuous Dynamical Systems - A 2014, 34(11): 4487-4513 doi: 10.3934/dcds.2014.34.4487
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.

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