Renewal theorems for a class of processes with dependent interarrival times and applications in geometry

Renewal theorems are developed for point processes with interarrival times $W_n=\xi(X_{n+1}X_n\cdots)$, where $(X_n)_{n\in\mathbb Z}$ is a stochastic process with finite state space $\Sigma$ and $\xi\colon\Sigma_A\to\mathbb R$ is a H\"older continuous function on a subset $\Sigma_A\subset\Sigma^{\mathbb N}$. The theorems developed here unify and generalise the key renewal theorem for discrete measures and Lalley's renewal theorem for counting measures in symbolic dynamics. Moreover, they capture aspects of Markov renewal theory. The new renewal theorems allow for direct applications to problems in fractal and hyperbolic geometry; for instance, results on the Minkowski measurability of self-conformal sets are deduced. Indeed, these geometric problems motivated the development of the renewal theorems.Comment: 2 figure

Minkowski content and fractal Euler characteristic for conformal graph directed systems

We study the (local) Minkowski content and the (local) fractal Euler characteristic of limit sets $F\subset\mathbb R$ of conformal graph directed systems (cGDS) $\Phi$. For the local quantities we prove that the logarithmic Ces\`aro averages always exist and are constant multiples of the $\delta$-conformal measure. If $\Phi$ is non-lattice, then also the non-average local quantities exist and coincide with their respective average versions. When the conformal contractions of $\Phi$ are analytic, the local versions exist if and only if $\Phi$ is non-lattice. For the non-local quantities the above results in particular imply that limit sets of Fuchsian groups of Schottky type are Minkowski measurable, proving a conjecture of Lapidus from 1993. Further, when the contractions of the cGDS are similarities, we obtain that the Minkowski content and the fractal Euler characteristic of $F$ exist if and only if $\Phi$ is non-lattice, generalising earlier results by Falconer, Gatzouras, Lapidus and van Frankenhuijsen for non-degenerate self-similar subsets of $\mathbb R$ that satisfy the open set condition.Comment: 34 page

Minkowski Content and local Minkowski Content for a class of self-conformal sets

We investigate (local) Minkowski measurability of $\mathcal C^{1+\alpha}$ images of self-similar sets. We show that (local) Minkowski measurability of a self-similar set $K$ implies (local) Minkowski measurability of its image $F$ and provide an explicit formula for the (local) Minkowski content of $F$ in this case. A counterexample is presented which shows that the converse is not necessarily true. That is, $F$ can be Minkowski measurable although $K$ is not. However, we obtain that an average version of the (local) Minkowski content of both $K$ and $F$ always exists and also provide an explicit formula for the relation between the (local) average Minkowski contents of $K$ and $F$.Comment: The final publication is available at http://www.springerlink.co

Fractal Curvature Measures and Minkowski Content for Limit Sets of Conformal Function Systems

We characterise fractal sets arising from conformal iterated function systems and conformal graph directed Markov systems for which the Minkowski content and the fractal curvature measures exist. With this, we generalise studies that have been carried out for invariant sets of iterated function systems consisting of similarities

Lattice-type self-similar sets with pluriphase generators fail to be Minkowski measurable

A long-standing conjecture of Lapidus claims that under certain conditions, self-similar fractal sets fail to be Minkowski measurable if and only if they are of lattice type. The theorem was established for fractal subsets of $\mathbb{R}$ by Falconer, Lapidus and v.~Frankenhuijsen, and the forward direction was shown for fractal subsets of $\mathbb{R}^d$, $d \geq 2$, by Gatzouras. Since then, much effort has been made to prove the converse. In this paper, we prove a partial converse by means of renewal theory. Our proof allows us to recover several previous results in this regard, but is much shorter and extends to a more general setting; several technical conditions appearing in previous versions of this result have now been removed.Comment: 20 pages, 6 figure

On bounds for the remainder term of counting functions of the Neumann Laplacian on domains with fractal boundary

We provide a new constructive method for obtaining explicit remainder estimates of eigenvalue counting functions of Neumann Laplacians on domains with fractal boundary. This is done by establishing estimates for first non-trivial eigenvalues through Rayleigh quotients. A main focus lies on domains whose boundary can locally be represented as a limit set of an IFS, with the classic Koch snowflake and certain Rohde snowflakes being prototypical examples, to which the new method is applied. Central to our approach is the construction of a novel foliation of the domain near its boundary.Comment: 22 pages, 9 figure

Eigenvalue counting functions and parallel volumes for examples of fractal sprays generated by the Koch snowflake

We apply recent results by the authors to obtain bounds on remainder terms of the Dirichlet Laplace eigenvalue counting function for domains that can be realised as countable disjoint unions of scaled Koch snowflakes. Moreover we compare the resulting exponents to the exponents in the asymptotic expansion of the domain's inner parallel volume.Comment: Added details, fixed typos. 15 pages, 10 figure