arXiv subjects
Publications and source records attributed to Johannes Jaerisch.
We give a short new proof that for each non-elementary Kleinian group $Γ$, the exponent of convergence of an arbitrary non-trivial normal subgroup is bounded below by half of the exponent of convergence of $Γ$, and that strict inequality holds if $Γ$ is of divergence type.
We characterise amenability of a countable group in terms of the spectral radius of the Perron-Frobenius operator associated to a group extension of a countable Markov shift and a Hölder continuous potential. This extends a result of Day for random walks and recent work of Stadlbauer for dynamical systems. Moreover, we show that, if the potential satisfies a symmetry condition with respect to the group extension, then the logarithm of the spectral radius of the Perron-Frobenius operator is given by the Gurevic pressure of the potential.
We investigate the thermodynamic formalism for recurrent potentials on group extensions of countable Markov shifts. Our main result characterises recurrent potentials depending only on the base space, in terms of the existence of a conservative product measure and a homomorphism from the group into the multiplicative group of real numbers. We deduce that, for a recurrent potential depending only on the base space, the group is necessarily amenable. Moreover, we give equivalent conditions for the base pressure and the skew product pressure to coincide. Finally, we apply our results to analyse the Poincaré series of Kleinian groups and the cogrowth of group presentations.
For a normal subgroup $N$ of the free group $\F_d$ with at least two generators we introduce the radial limit set $\Lr(N,Φ)$ of $N$ with respect to a graph directed Markov system $Φ$ associated to $\F_d$. These sets are shown to provide fractal models of radial limit sets of normal subgroups of Kleinian groups of Schottky type. Our main result states that if $Φ$ is symmetric and linear, then we have that $\dim_{H}(\Lr(N,Φ))=\dim_{H} \Lr(\F_d,Φ))$ if and only if the quotient group $\F_d /N$ is amenable, where $\dim_{H}$ denotes the Hausdorff dimension. This extends a result of Brooks for normal subgroups of Kleinian groups to a large class of fractal sets. Moreover, we show that if $\F_d /N$ is non-amenable then $\dim_{H}(\Lr(N,Φ))>\dim_{H}(\Lr(\F_d,Φ))/2$, which extends results by Falk and Stratmann and by Roblin.
In this paper we establish a Fréchet law for maximal cuspidal windings of the geodesic flow on a Riemannian surface associated with an arbitrary finitely generated, essentially free Fuchsian group with parabolic elements. This result extends previous work by Galambos and Dolgopyat and is obtained by applying Extreme Value Theory. Subsequently, we show that this law gives rise to an Erdös-Philipp law and to various generalised Khintchine-type results for maximal cuspidal windings. These results strengthen previous results by Sullivan, Stratmann and Velani for Kleinian groups, and extend earlier work by Philipp on continued fractions, which was inspired by a conjecture of Erdös.
We introduce the notion of induced topological pressure for countable state Markov shifts with respect to a non-negative scaling function and an arbitrary subset of finite words. Firstly, the scaling function allows a direct access to important thermodynamical quantities, which are usually given only implicitly by certain identities involving the classically defined pressure. In this context we generalise Savchenko's definition of entropy for special flows to a corresponding notion of topological pressure and show that this new notion coincides with the induced pressure for a large class of Hölder continuous height functions not necessarily bounded away from zero. Secondly, the dependence on the subset of words gives rise to interesting new results connecting the Gurevi{\vc} and the classical pressure with exhausting principles for a large class of Markov shifts. In this context we consider dynamical group extentions to demonstrate that our new approach provides a useful tool to characterise amenability of the underlying group structure.
We investigate multifractal regularity for infinite conformal iterated function systems (cIFS). That is we determine to what extent the multifractal spectrum depends continuously on the cIFS and its thermodynamic potential. For this we introduce the notion of regular convergence for families of cIFS not necessarily sharing the same index set, which guarantees the convergence of the multifractal spectra on the interior of their domain. In particular, we obtain an Exhausting Principle for infinite cIFS allowing us to carry over results for finite to infinite systems, and in this way to establish a multifractal analysis without the usual regularity conditions. Finally, we discuss the connections to the $λ$-topology introduced by Roy and Urbas{ń}ki.
To compare continued fraction digits with the denominators of the corresponding approximants we introduce the arithmetic-geometric scaling. We will completely determine its multifractal spectrum by means of a number theoretical free energy function and show that the Hausdorff dimension of sets consisting of irrationals with the same scaling exponent coincides with the Legendre transform of this free energy function. Furthermore, we identify the asymptotic of the local behaviour of the spectrum at the right boundary point and discuss a connection to the set of irrationals with continued fraction digits exceeding a given number which tends to infinity.