arXiv subjects

Johannes Jaerisch

Publications and source records attributed to Johannes Jaerisch.

At least 19 recordsLinked to original sources

arXiv2026

Amenable graphs and the spectral radius of extensions of Markov maps

We discuss relations between the amenability of a graph and spectral properties of a random walk driven by a dynamical system. In order to include graphs which are not locally compact, we introduce the concept of amenability of weighted graphs, which generalises the usual notion as the new definition is shown to be equivalent to Folner's condition. As a first result, we obtain the following generalisation of Kesten's amenability criterion to graphs and non-independent increments: If the random walk is driven by a full-branched Gibbs-Markov map, the graph is amenable with respect to the weight induced by the random walk if and only if the spectral radius of the associated Markov operator is equal to one. By employing inducing schemes, one then obtains criteria for amenability through Markov maps with less regularity. We conclude the paper with the following applications to Schreier graphs. If the random walk is driven by a uniformly expanding map with non-Markovian increments or a Sinai billiard, then, under certain conditions, the Schreier graph is amenable if the probability of a return in time n does not decay exponentially in n. Furthermore, in the context of geometrically finite Kleinian groups, one obtains a version of Brooks's amenability criterion for not necessarily normal subgroups.

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arXiv2026

Equality of Hölder exponents for distribution functions of Gibbs measures

Pointwise Hölder exponents describe the degree of regularity of a function near a point. For a function $f:\mathbb{R}\to\mathbb{R}$, a number $α>0$ and a point $t_0\in\mathbb{R}$, write $f\in C^α(t_0)$ if there exist a constant $C>0$, a number $h>0$ and a polynomial $P$ of degree less than $α$ such that \[ |f(t)-P(t-t_0)|\leq C|t-t_0|^α\qquad\mbox{for all $t\in (t_0-h,t_0+h)$}. \] The pointwise Hölder exponent of $f$ at $t_0$ is the number \[ α_f(t_0):=\sup\{α>0: f\in C^α(t_0)\}. \] A simpler quantity, also frequently called pointwise Hölder exponent in the mathematical literature, is the number \[ \tildeα_f(t_0):=\sup\{α>0: f\in \tilde{C}^α(t_0)\}, \] where $f\in \tilde{C}^α(t_0)$ means that there exist $C>0$ and $h>0$ such that $|f(t)-f(t_0)|\leq C|t-t_0|^α$ for all $t\in (t_0-h,t_0+h)$. Clearly $α_f(t)\geq \tildeα_f(t)$, but strict inequality is possible and in fact common. In this paper we consider the case when $f=F_μ$ is the distribution function of a Gibbs measure $μ$ associated with an arbitrary Hölder continuous potential $ψ$ on a self-conformal set, and show that, under a very mild condition on $ψ$, $α_f(t)=\tildeα_f(t)$ for all $t$. As a consequence, we deduce that the pointwise Hölder spectrum of $f$ satisfies the multifractal formalism. As an application, we derive the pointwise Hölder spectrum of conjugacy maps between expanding piecewise $\mathcal{C}^{1+ε}$ maps of an interval.

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arXiv2026

Dimension gap and phase transition for one-dimensional random walks with reflective boundary

We study $\mathbb Z$- and $\mathbb N$-extensions of interval maps with at most countably many full branches modelling one-dimensional random walks without and with a reflective boundary. We analyse the associated Gurevich pressure and explore the relations governing these two cases. For such extensions, we obtain variational formulae for the Gurevich pressure that depend only on the base system. As a consequence, we characterise the systems with a dimension gap and, in the presence of a reflective boundary, provide general conditions in terms of asymptotic covariances for a second order phase transition. As a by-product, we derive a variational formula for the spectral radius of infinite Hessenberg matrices.

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arXiv2024

Bowen's formula for a rational graph-directed Markov system

We establish Bowen's formula for the Julia set of a non-elementary, expanding, irreducible and aperiodic rational graph-directed Markov system satisfying the backward separating condition. Towards this end, we shall prove that the associated skew product map is topologically exact on the skew product Julia set, and satisfies the density of repelling periodic points. Moreover, we give a criterion for expandingness in terms of hyperbolicity.

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arXiv2023

Large deviations of homological growth rates for hyperbolic surfaces

We perform a large deviations analysis of homological growth rates of oriented geodesics on hyperbolic surfaces. For surfaces uniformized by a wide class of Fuchsian groups of the first kind, we prove the existence of the rate function which estimates exponential probabilities with which the homological growth rates stay away from the mean value. The rate function is given in terms of the multifractal dimension spectrum described in our earlier result [arXiv:2204.08907]. We also establish an Erdős-Rényi law, and refined large deviations upper bounds.

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arXiv2023

Multifractal analysis of homological growth rates for hyperbolic surfaces

We perform a multifractal analysis of homological growth rates of oriented geodesics on hyperbolic surfaces. Our main result provides a formula for the Hausdorff dimension of level sets of prescribed growth rates in terms of a generalized Poincaré exponent of the Fuchsian group. We employ symbolic dynamics developed by Bowen and Series, ergodic theory and thermodynamic formalism to prove the analyticity of the dimension spectrum.

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arXiv2020

Mixed multifractal spectra of Birkhoff averages for non-uniformly expanding one-dimensional Markov maps with countably many branches

For a Markov map of an interval or the circle with countably many branches and finitely many neutral periodic points, we establish conditional variational formulas for the mixed multifractal spectra of Birkhoff averages of countably many observables, in terms of the Hausdorff dimension of invariant probability measures. Using our results, we are able to exhibit new fractal-geometric results for backward continued fraction expansions of real numbers, answering in particular a question of Pollicott. Moreover, we establish formulas for multi-cusp winding spectra for the Bowen-Series maps associated with finitely generated free Fuchsian groups with parabolic elements.

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arXiv2020

Spectral gap property for random dynamics on the real line and multifractal analysis of generalised Takagi functions

We consider the random iteration of finitely many expanding $\mathcal{C}^{1+ε}$ diffeomorphisms on the real line without a common fixed point. We derive the spectral gap property of the associated transition operator acting on Hölder spaces. As an application we introduce generalised Takagi functions on the real line and we perform a complete multifractal analysis of the pointwise Hölder exponents of these functions.

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arXiv2019

Thermodynamic formalism for transient dynamics on the real line

We develop a new thermodynamic formalism to investigate the transient behaviour of maps on the real line which are skew-periodic $\mathbb{Z}$-extensions of expanding interval maps. Our main focus lies in the dimensional analysis of the recurrent and transient sets as well as in determining the whole dimension spectrum with respect to $α$-escaping sets. Our results provide a one-dimensional model for the phenomenon of a dimension gap occurring for limit sets of Kleinian groups. In particular, we show that a dimension gap occurs if and only if we have non-zero drift and we are able to precisely quantify its width as an application of our new formalism.

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arXiv2019

Normalizer, divergence type and Patterson measure for discrete groups of the Gromov hyperbolic space

For a non-elementary discrete isometry group $G$ of divergence type acting on a proper geodesic $δ$-hyperbolic space, we prove that its Patterson measure is quasi-invariant under the normalizer of $G$. As applications of this result, we have: (1) under a minor assumption, such a discrete group $G$ admits no proper conjugation, that is, if the conjugate of $G$ is contained in $G$, then it coincides with $G$; (2) the critical exponent of any non-elementary normal subgroup of $G$ is strictly greater than half of that for $G$.

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arXiv2019

Multifractal Formalism for generalised local dimension spectra of Gibbs measures on the real line

We refine the multifractal formalism for the local dimension of a Gibbs measure $μ$ supported on the attractor $Λ$ of a conformal iterated functions system on the real line. Namely, for given $α\in \mathbb{R}$, we establish the formalism for the Hausdorff dimension of level sets of points $x\inΛ$ for which the $μ$-measure of a ball of radius $r_{n}$ centered at $x$ obeys a power law $r_{n}{}^α$, for a sequence $r_{n}\rightarrow0$. This allows us to investigate the Hölder regularity of various fractal functions, such as distribution functions and conjugacy maps associated with conformal iterated function systems.

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arXiv2018

A multifractal analysis for cuspidal windings on hyperbolic surfaces

In this paper we investigate the multifractal decomposition of the limit set of a finitely generated, free Fuchsian group with respect to the mean cusp winding number. We will completely determine its multifractal spectrum by means of a certain free energy function and show that the Hausdorff dimension of sets consisting of limit points with the same scaling exponent coincides with the Legendre transform of this free energy function. As a by-product we generalise previously obtained results on the multifractal formalism for infinite iterated function systems to the setting of infinite graph directed Markov systems.

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arXiv2018

Weighted cogrowth formula for free groups

We investigate the relationship between geometric, analytic and probabilistic indices for quotients of the Cayley graph of the free group ${\rm Cay}(F_n)$ endowed with variable edge lengths, by an arbitrary subgroup $G$ of $F_n$. Our main result, which generalizes Grigorchuk's cogrowth formula to variable edge lengths, provides a formula relating the bottom of the spectrum of weighted Laplacian on $G \backslash {\rm Cay}(F_n)$ to the Poincaré exponent of $G$. Our main tool is the Patterson-Sullivan theory for Cayley graphs with variable edge lengths.

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arXiv2017

Pointwise Hölder Exponents of the Complex Analogues of the Takagi Function in Random Complex Dynamics

We investigate the Hölder regularity of the function $T$ of the probability of tending to one minimal set, the partial derivatives of $T$ with respect to the probability parameters, which can be regarded as complex analogues of the Takagi function, and the higher partial derivatives $C$ of $T.$ Our main result gives a dynamical description of the pointwise Hölder exponents of $T$ and $C$, which allows us to determine the spectrum of pointwise Hölder exponents by employing the multifractal formalism in ergodic theory. Also, we prove that the bottom of the spectrum $α_{-}$ is strictly less than $1$, which allows us to show that the averaged system acts chaotically on the Banach space $C^{α}$ of $α$- Hölder continuous functions for every $α\in (α_{-},1)$, though the averaged system behaves very mildly (e.g. we have spectral gaps) on $C^{β}$ for small $β>0.$

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arXiv2017

Dynamics of infinitely generated nicely expanding rational semigroups and the inducing method

We investigate the dynamics of semigroups of rational maps on the Riemann sphere. To establish a fractal theory of the Julia sets of infinitely generated semigroups of rational maps, we introduce a new class of semigroups which we call nicely expanding rational semigroups. More precisely, we prove Bowen's formula for the Hausdorff dimension of the pre-Julia sets, which we also introduce in this paper. We apply our results to the study of the Julia sets of non-hyperbolic rational semigroups. For these results, we do not assume the cone condition, which has been assumed in the study of infinite contracting iterated function systems. Similarly, we show that Bowen's formula holds for the limit set of a contracting conformal iterated function system without the cone condition.

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arXiv2015

Growth and cogrowth of normal subgroups of a free group

We give a sufficient condition for a sequence of normal subgroups of a free group to have the property that both, their growths tend to the upper bound and their cogrowths tend to the lower bound. The condition is represented by planarity of the quotient graphs of the tree.

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arXiv2015

Multifractal formalism for expanding rational semigroups and random complex dynamical systems

We consider the multifractal formalism for the dynamics of semigroups of rational maps on the Riemann sphere and random complex dynamical systems. We elaborate a multifractal analysis of level sets given by quotients of Birkhoff sums with respect to the skew product associated with a semigroup of rational maps. Applying these results, we perform a multifractal analysis of the Hölder regularity of limit state functions of random complex dynamical systems.

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arXiv2013

Conformal Fractals for Normal Subgroups of Free Groups

We investigate subsets of a multifractal decomposition of the limit set of a conformal graph directed Markov system, which is constructed from the Cayley graph of a free group with at least two generators. The subsets we consider are parametrised by a normal subgroup $N$ of the free group and mimic the radial limit set of a Kleinian group. Our main results show that, regarding the Hausdorff dimension of these sets, various results for Kleinian groups can be generalised. Namely, under certain natural symmetry assumptions on the multifractal decomposition, we prove that, for a subset parametrised by $N$, the Hausdorff dimension is maximal if and only if $\F_{d}/N$ is amenable and that the dimension is greater than half of the maximal value. We also give a criterion for amenability via the divergence of the Poincaré series of $N$. Our results are applied to the Lyapunov spectrum for normal subgroups of Kleinian groups of Schottky type.

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