We give a description of the level sets in the higher dimensional multifractal formalism for infinite conformal graph directed Markov systems. If these systems possess a certain degree of regularity this description is complete in the sense that we identify all values with non-empty level sets and determine their Hausdorff dimension. This result is also partially new for the finite alphabet case.
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…
We introduce multifractal pressure and dynamical multifractal zeta-functions providing precise information of a very general class of multifractal spectra, including, for example, the fine multifractal spectra of graph-directed self-conformal measures and the fine multifractal spectra of ergodic Birkhoff averages of continuous functions on graph-directed self-conformal sets.
We derive the multifractal analysis of the conformal measure (or equivalently, the invariant measure) associated to a family of weights imposed upon a (multi-dimensional) graph directed Markov system (GDMS) using balls as the filtration. This analysis is done over a large subset of the limit set. In particular, it coincides with the limit set when the GDMS under scrutiny satisfies a boundary separation condition. It also applies to more general situations such as real or complex continued fractions.
Multifractal analysis refers to the study of the local properties of measures and functions, and consists of two parts: the fine multifractal theory and the coarse multifractal theory. The fine and the coarse theory are linked by a web of conjectures known collectively as the Multifractal Formalism. Very roughly speaking the Multifractal Formalism says that the multifractal spectrum from fine theory equals the Legendre transform of the Renyi dimensions from the coarse theory. Recently {\it fine} multifractal zeta-functions, i.e. multifractal zeta-functions designed to produce detailed informat…
We consider the multifractal analysis for Birkhoff averages of continuous potentials on a class of non-conformal repellers corresponding to the self-affine limit sets studied by Lalley and Gatzouras. A conditional variational principle is given for the Hausdorff dimension of the set of points for which the Birkhoff averages converge to a given value. This extends a result of Barral and Mensi to certain non-conformal maps with a measure dependent Lyapunov exponent.
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 iterate…
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 di…
We conduct the multifractal analysis of the level sets of the asymptotic behavior of almost-additive continuous potentials $(ϕ_n)_{n=1}^\infty$ on a topologically mixing subshift of finite type $X$ endowed itself with a metric associated with such a potential. We work without bounded distorsion property assumption. We express the whole Hausdorff spectrum in terms of a conditional variational principle, as well as a new large deviations principle. Our approach provides a new description of the structure of the spectrum in terms of {\it weak} concavity. Another new point is that we consider sets…
We present the main concepts and results for Graph Directed Markov Systems that have a finitely irreducible incidence matrix. We then see how these results change when the incidence matrix is not assumed to be finitely irreducible.
In this paper, we consider two dynamical systems associated to the nearest integer continued fraction, and show that both of them have full Hausdorff dimension spectrum.
We consider a generalisation of the self-affine iterated function systems of Lalley and Gatzouras by allowing for a countable infinity of non-conformal contractions. It is shown that the Hausdorff dimension of the limit set is equal to the supremum of the dimensions of compactly supported ergodic measures. In addition we consider the multifractal analysis for countable families of potentials. We obtain a conditional variational principle for the level sets.
In this paper we introduce and develop the theory of non-autonomous graph directed Markov systems which is a generalization of the theory of conformal graph directed Markov systems of Mauldin and Urbański, first presented in their book, and the theory of non-autonomous conformal iterated function systems set forth by Rempe-Gillen and Urbański. We exhibit several large classes of functions for which Bowen's formula for Hausdorff dimension holds. In particular we consider weakly balanced finite systems, where we have some control over the growth of the derivatives, and ascending systems. Our res…
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 parametrise…
Universal projection theorems with applications to multifractal analysis and the dimension of every ergodic measure on self-conformal sets simultaneously
The quantization dimension function for an $F$-conformal measure $m_F$ generated by an infinite conformal iterated function system satisfying the strong open set condition and by a summable Hölder family of functions is expressed by a simple formula involving the temperature function of the system. The temperature function is commonly used to perform the multifractal analysis, in our context of the measure $m_F$. The result in this paper extends a similar result of Lindsay and Mauldin established for finite conformal iterated function systems [Nonlinearity 15 (2002)].
We develop a comprehensive theory of conformal graph directed Markov systems in the non-Riemannian setting of Carnot groups equipped with a sub-Riemannian metric. In particular, we develop the thermodynamic formalism and show that, under natural hypotheses, the limit set of an Carnot conformal GDMS has Hausdorff dimension given by Bowen's parameter. We illustrate our results for a variety of examples of both linear and nonlinear iterated function systems and graph directed Markov systems in such sub-Riemannian spaces. These include the Heisenberg continued fractions introduced by Lukyanenko an…
The aim of the present paper is twofold. We study directed porosity in connection with conformal iterated function systems (CIFS) and with singular integrals. We prove that limit sets of finite CIFS are porous in a stronger sense than already known. Furthermore we use directed porosity to establish that truncated singular integral operators, with respect to general Radon measures $μ$ and kernels $K$, converge weakly in some dense subspaces of $L^2(μ)$ when the support of $μ$ belongs to a broad family of sets. This class contains many fractal sets like CIFS's limit sets.
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) $Φ$. For the local quantities we prove that the logarithmic Cesàro averages always exist and are constant multiples of the $δ$-conformal measure. If $Φ$ is non-lattice, then also the non-average local quantities exist and coincide with their respective average versions. When the conformal contractions of $Φ$ are analytic, the local versions exist if and only if $Φ$ is non-lattice. For the non-local quantities the above results in parti…
We give conditions for the existence of the Minkowski content of limit sets stemming from infinite conformal graph directed systems. As an application we obtain Minkowski measurability of Apollonian gaskets, provide explicit formulae of the Minkowski content, and prove the analytic dependence on the initial circles. Further, we are able to link the fractal Euler characteristic, as well as the Minkowski content, of Apollonian gaskets with the asymptotic behaviour of the circle counting function studied by Kontorovich and Oh. These results lead to a new interpretation and an alternative formula …
In this paper we study the dimension spectrum of general conformal graph directed Markov systems modeled by countable state symbolic subshifts of finite type. We perform a comprehensive study of the dimension spectrum addressing questions regarding its size and topological structure. As a corollary we obtain that the dimension spectrum of infinite conformal iterated function systems is compact and perfect. On the way we revisit the role of the parameter $θ$ in graph directed Markov systems and we show that new phenomena arise. We also establish topological pressure estimates for subsystems in …
We investigate the Hausdorff measure and content on a class of quasi self-similar sets that include, for example, graph-directed and sub self-similar and self-conformal sets. We show that any Hausdorff measurable subset of such a set has comparable Hausdorff measure and Hausdorff content. In particular, this proves that graph-directed and sub self-conformal sets with positive Hausdorff measure are Ahlfors regular, irrespective of separation conditions. When restricting to self-conformal subsets of the real line with Hausdorff dimension strictly less than one, we additionally show that the weak…
The study of escape rates for a ball in a dynamical systems has been much studied. Understanding the asymptotic behavior of the escape rate as the radius of the ball tends to zero is an especially subtle problem. In the case of hyperbolic conformal systems this has been addressed by various authors. In this paper we consider a far more general realm of conformal maps where the analysis is correspondingly more complicated. We prove the existence of escape rates and calculate them in the context of countable alphabets, either finite or infinite, uniformly contracting conformal graph directed Mar…
