arXiv subjects

Sascha Troscheit

Publications and source records attributed to Sascha Troscheit.

31 records · Page 2Linked to original sources

arXiv2019

Exact Hausdorff and packing measures for random self-similar code-trees with necks

Random code-trees with necks were introduced recently to generalise the notion of $V$-variable and random homogeneous sets. While it is known that the Hausdorff and packing dimensions coincide irrespective of overlaps, their exact Hausdorff and packing measure has so far been largely ignored. In this article we consider the general question of an appropriate gauge function for positive and finite Hausdorff and packing measure. We first survey the current state of knowledge and establish some bounds on these gauge functions. We then show that self-similar code-trees do not admit a gauge functions that simultaneously give positive and finite Hausdorff measure almost surely. This surprising result is in stark contrast to the random recursive model and sheds some light on the question of whether $V$-variable sets interpolate between random homogeneous and random recursive sets. We conclude by discussing implications of our results.

math.MG↗

arXiv2018

Lower Assouad Dimension of Measures and Regularity

In analogy with the lower Assouad dimensions of a set, we study the lower Assouad dimensions of a measure. As with the upper Assouad dimensions, the lower Assouad dimensions of a measure provide information about the extreme local behaviour of the measure. We study the connection with other dimensions and with regularity properties. In particular, the quasi-lower Assouad dimension is dominated by the infimum of the measure's lower local dimensions. Although strict inequality is possible in general, equality holds for the class of self-similar measures of finite type. This class includes all self-similar, equicontractive measures satisfying the open set condition, as well as certain `overlapping' self-similar measures, such as Bernoulli convolutions with contraction factors that are inverses of Pisot numbers. We give lower bounds for the lower Assouad dimension for measures arising from a Moran construction, prove that self-affine measures are uniformly perfect and have positive lower Assouad dimension, prove that the Assouad spectrum of a measure converges to its quasi-Assouad dimension and show that coincidence of the upper and lower Assouad dimension of a measure does not imply that the measure is $s$-regular.

math.MG↗

arXiv2018

Quasi-doubling of self-similar measures with overlaps

The Assouad and quasi-Assouad dimensions of a metric space provide information about the extreme local geometric nature of the set. The Assouad dimension of a set has a measure theoretic analogue, which is also known as the upper regularity dimension. One reason for the interest in this notion is that a measure has finite Assouad dimension if and only if it is doubling. Motivated by recent progress on both the Assouad dimension of measures that satisfy a strong separation condition and the quasi-Assouad dimension of metric spaces, we introduce the notion of the quasi-Assouad dimension of a measure. As with sets, the quasi-Assouad dimension of a measure is dominated by its Assouad dimension. It dominates both the quasi-Assouad dimension of its support and the supremal local dimension of the measure, with strict inequalities possible in all cases. Our main focus is on self-similar measures in $\mathbb{R}$ whose support is an interval and which may have `overlaps'. For measures that satisfy a weaker condition than the weak separation condition we prove that finite quasi-Assouad dimension is equivalent to quasi-doubling of the measure, a strictly less restrictive property than doubling. Further, we exhibit a large class of such measures for which the quasi-Assouad dimension coincides with the maximum of the local dimension at the endpoints of the support. This class includes all regular, equicontractive self-similar measures satisfying the weak separation condition, such as convolutions of uniform Cantor measures with integer ratio of dissection. Other properties of this dimension are also established and many examples are given.

math.MG↗

arXiv2018

The Assouad spectrum of random self-affine carpets

We derive the almost sure Assouad spectrum and quasi-Assouad dimension of random self-affine Bedford-McMullen carpets. Previous work has revealed that the (related) Assouad dimension is not sufficiently sensitive to distinguish between subtle changes in the random model, since it tends to be almost surely `as large as possible' (a deterministic quantity). This has been verified in conformal and non-conformal settings. In the conformal setting, the Assouad spectrum and quasi-Assouad dimension behave rather differently, tending to almost surely coincide with the upper box dimension. Here we investigate the non-conformal setting and find that the Assouad spectrum and quasi-Assouad dimension generally do not coincide with the box dimension or Assouad dimension. We provide examples highlighting the subtle differences between these notions. Our proofs combine deterministic covering techniques with suitably adapted Chernoff estimates and Borel-Cantelli type arguments.

math.DS↗

arXiv2018

The Assouad spectrum and the quasi-Assouad dimension: a tale of two spectra

We consider the Assouad spectrum, introduced by Fraser and Yu, along with a natural variant that we call the `upper Assouad spectrum'. These spectra are designed to interpolate between the upper box-counting and Assouad dimensions. It is known that the Assouad spectrum approaches the upper box-counting dimension at the left hand side of its domain, but does not necessarily approach the Assouad dimension on the right. Here we show that it necessarily approaches the \emph{quasi-Assouad dimension} at the right hand side of its domain. We further show that the upper Assouad spectrum can be expressed in terms of the Assouad spectrum, thus motivating the definition used by Fraser-Yu. We also provide a large family of examples demonstrating new phenomena relating to the form of the Assouad spectrum. For example, we prove that it can be strictly concave, exhibit phase transitions of any order, and need not be piecewise differentiable.

math.CA↗

arXiv2017

Local dimensions of random homogeneous self-similar measures: strong separation and finite type

We study the multifractal analysis of self-similar measures arising from random homogeneous iterated function systems. Under the assumption of the uniform strong separation condition, we see that this analysis parallels that of the deterministic case. The overlapping case is more complicated; we introduce the notion of finite type for random homogeneous iterated function systems and give a formula for the local dimensions of finite type, regular, random homogeneous self-similar measures in terms of Lyapunov exponents of certain transition matrices. We show that almost all points with respect to this measure are described by a distinguished subset called the essential class, and that the dimension of the support can be computed almost surely from knowledge of this essential class. For a special subcase, that we call commuting, we prove that the set of attainable local dimensions is almost surely a closed interval. Particular examples of such random measures are analyzed in more detail.

math.DS↗

arXiv2017

The quasi-Assouad dimension for stochastically self-similar sets

The class of stochastically self-similar sets contains many famous examples of random sets, e.g. Mandelbrot percolation and general fractal percolation. Under the assumption of the uniform open set condition and some mild assumptions on the iterated function systems used, we show that the quasi-Assouad dimension of self-similar random recursive sets is almost surely equal to the almost sure Hausdorff dimension of the set. We further comment on random homogeneous and $V$-variable sets and the removal of overlap conditions.

math.MG↗

arXiv2017

The box dimension of random box-like self-affine sets

In this paper we study two random analogues of the box-like self-affine attractors introduced by Fraser, itself an extension of Sierpiński carpets. We determine the almost sure box-counting dimension for the homogeneous random case ($1$-variable random), and give a sufficient condition for the almost sure box dimension to be the expectation of the box dimensions of the deterministic attractors. Furthermore we find the almost sure box-counting dimension of the random recursive model ($\infty$-variable), which includes affine fractal percolation.

math.MG↗

arXiv2017

On the dimensions of attractors of random self-similar graph directed iterated function systems

In this paper we propose a new model of random graph directed fractals that extends the current well-known model of random graph directed iterated function systems, $V$-variable attractors, and fractal and Mandelbrot percolation. We study its dimensional properties for similarities with and without overlaps. In particular we show that for the two classes of $1$-variable and $\infty$-variable random graph directed attractors we introduce, the Hausdorff and upper box counting dimension coincide almost surely, irrespective of overlap. Under the additional assumption of the uniform strong separation condition we give an expression for the almost sure Hausdorff and Assouad dimension.

math.MG↗

arXiv2017

Random ubiquitous transformation semigroups

A smallest generating set of a semigroup is a generating set of the smallest cardinality. Similarly, an irredundant generating set $X$ is a generating set such that no proper subset of $X$ is also a generating set. A semigroup $S$ is ubiquitous if every irredundant generating set of $S$ is of the same cardinality. We are motivated by a naïve algorithm to find a small generating set for a semigroup, which in practice often outputs a smallest generating set. We give a sufficient condition for a transformation semigroup to be ubiquitous and show that a transformation semigroup generated by $k$ randomly chosen transformations asymptoticly satisfies the sufficient condition. Finally, we show that under this condition the output of the previously mentioned naïve algorithm is irredundant.

math.GR↗

arXiv2017

The Mass Transference Principle: Ten Years On

In this article we discuss the Mass Transference Principle due to Beresnevich and Velani and survey several generalisations and variants, both deterministic and random. Using a Hausdorff measure analogue of the inhomogeneous Khintchine-Groshev Theorem, proved recently via an extension of the Mass Transference Principle to systems of linear forms, we give an alternative proof of a general inhomogeneous Jarn\'ık-Besicovitch Theorem which was originally proved by Levesley. We additionally show that without monotonicity Levesley's theorem no longer holds in general. Thereafter, we discuss recent advances by Wang, Wu and Xu towards mass transference principles where one transitions from $\limsup$ sets defined by balls to $\limsup$ sets defined by rectangles (rather than from "balls to balls" as is the case in the original Mass Transference Principle). Furthermore, we consider mass transference principles for transitioning from rectangles to rectangles and extend known results using a slicing technique. We end this article with a brief survey of random analogues of the Mass Transference Principle.

math.NT↗

arXiv2015

The Assouad dimension of randomly generated fractals

We consider several different models for generating random fractals including random self-similar sets, random self-affine carpets, and fractal percolation. In each setting we compute either the \emph{almost sure} or the \emph{Baire typical} Assouad dimension and consider some illustrative examples. Our results reveal a common phenomenon in all of our models: the Assouad dimension of a randomly generated fractal is generically as big as possible and does not depend on the measure theoretic or topological structure of the sample space. This is in stark contrast to the other commonly studied notions of dimension like the Hausdorff or packing dimension.

math.MG↗

arXiv2013

Hölder differentiability of self-conformal devil's staircases

In this paper we consider the probability distribution function of a Gibbs measure supported on a self-conformal set given by an iterated function system (devil's staircase). We use thermodynamic multifractal formalism to calculate the Hausdorff dimension of the sets $S^α_{0}$, $S^α_{\infty}$ and $S^α$, the set of points at which this function has, respectively, Hölder derivative 0, $\infty$ or no derivative in the general sense. This extends recent work by Darst, Dekking, Falconer, Kesseböhmer and Stratmann and Yao, Zhang and Li.

math.DS↗