In a first part, using the recent measure classification results of Eskin--Lindenstrauss, we give a criterion to ensure a.s. equidistribution of empirical measures of an i.i.d. random walk on a homogeneous space $G/Γ$. Employing renewal and joint equidistribution arguments, this result is generalized in the second part to random walks with Markovian dependence. Finally, following a strategy of Simmons--Weiss, we apply these results to Diophantine approximation problems on fractals and show that almost every point with respect to Hausdorff measure on a graph directed self-similar set is of generic type, so in particular, well approximable.
We describe in this survey several results relating Fractal Geometry, Dynamical Systems and Diophantine Approximations, including a description of recent results related to geometrical properties of the classical Markov and Lagrange spectra and generalizations in Dynamical Systems and Differential Geometry.
We extend results of Y. Benoist and J.-F. Quint concerning random walks on homogeneous spaces of simple Lie groups to the case where the measure defining the random walk generates a semigroup which is not necessarily Zariski dense, but satisfies some expansion properties for the adjoint action. Using these dynamical results, we study diophantine properties of typical points on some self-similar fractals in R^d. As examples, we show that for any self-similar fractal K in R^d satisfying the open set condition (for instance any translate or dilate of Cantor's middle thirds set or of a Koch snowfl…
We exploit dynamical properties of diagonal actions to derive results in Diophantine approximations. In particular, we prove that the continued fraction expansion of almost any point on the middle third Cantor set (with respect to the natural measure) contains all finite patterns (hence is well approximable). Similarly, we show that for a variety of fractals in [0,1]^2, possessing some symmetry, almost any point is not Dirichlet improvable (hence is well approximable) and has property C (after Cassels). We then settle by similar methods a conjecture of M. Boshernitzan saying that there are no …
We consider improvements of Dirichlet's Theorem on space of matrices $M_{m,n}(R)$. It is shown that for a certain class of fractals $K\subset [0,1]^{mn}\subset M_{m,n}(R)$ of local maximal dimension Dirichlet's Theorem cannot be improved almost everywhere. This is shown using entropy and dynamics on homogeneous spaces of Lie groups.
Non-Newtonian calculus that starts with elementary non-Diophantine arithmetic operations of a Burgin type is applicable to all fractals whose cardinality is continuum. The resulting definitions of derivatives and integrals are simpler from what one finds in the more traditional literature of the subject, and they often work in the cases where the standard methods fail. As an illustration, we perform a Fourier transform of a real-valued function with Sierpiński-set domain. The resulting formalism is as simple as the usual undergraduate calculus.
This work addresses problems on simultaneous Diophantine approximation on fractals, motivated by a long standing problem of Mahler regarding Cantor's middle $1/3$ set. We obtain the first instances where a complete analogue of Khintchine's Theorem holds for fractal measures. Our results apply to fractals which are self-similar by a system of rational similarities of $\mathbb{R}^d$ (for any $d\geq 1$) and have sufficiently small Hausdorff co-dimension. A concrete example of such measures in the context of Mahler's problem is the Hausdorff measure on the "middle $1/5$ Cantor set"; i.e. the set o…
Walk dimension is an important conception in analysis of fractals. In this paper we prove that the walk dimension of a connected compact set possessing an Alfors regular measure is an invariant under Lipschitz transforms. As an application, we show some generalized Sierpiński gaskets are not Lipschitz equivalent.
We study the convergence of the parameter family of series $$V_{α,β}(t)=\sum_{p}p^{-α}\exp(2πi p^βt),\quad α,β\in \mathbb{R}_{>0},\; t \in [0,1)$$ defined over prime numbers $p$, and subsequently, their differentiability properties. The visible fractal nature of the graphs as a function of $α,β$ is analyzed in terms of Hölder continuity, self similarity and fractal dimension, backed with numerical results. We also discuss the link of this series to random walks and consequently, explore numerically its random properties.
For a given pcf self-similar fractal, a certain network (weighted graph) is constructed whose ideal boundary is (homeomorphic to) the fractal. This construction is the first representation of a connected self-similar fractal as the boundary of a reversible Markov chain (i.e., a simple random walk on a network). The boundary construction is effected using certain functions of finite energy which behave like bump functions on the boundary. The random walk is shown to converge to the boundary almost surely, with respect to the standard measure on its trajectory space.
Fix an irrational number $α$, and consider a random walk on the circle in which at each step one moves to $x+α$ or $x-α$ with probabilities $1/2, 1/2$ provided the current position is $x$. If an observable is given we can study a process called an additive functional of this random walk. One can formulate certain relations between the regularity of the observable and the Diophantine properties of $α$ implying the central limit theorem. It is proven here that for every Liouville angle there exists a smooth observable such that the central limit theorem fails. We construct also a Liouville angle…
We introduce new method of optimization for finding free parameters of affine iterated function systems (IFS), which are used for fractal approximation. We provide the comparison of effectiveness of fractal and quadratic types of approximation, which are based on a similar optimization scheme, on the various types of data: polynomial function, DNA primary sequence, price graph and graph of random walking.
We show that for any $ε<1$ and any $\mathcal{T}$ `drifting away from walls', Dirichlet's Theorem cannot be $ε$-improved along $\mathcal{T}$ for Lebesgue almost every system of linear forms $Y$ (see the paper for definitions). In the case $m = 1$ we also show that for a large class of measures $μ$ there is $ε_0>0$ such that for any drifting away from walls $\mathcal{T}$, any $ε<ε_0$, and for $μ$-almost every $Y$, Dirichlet's Theorem cannot be $ε$-improved along $\mathcal{T}$. These measures include natural measures on sufficiently regular smooth manifolds and fractals. Our results extend those …
Let $ρ$ be a probability measure on $\mathrm{SL}\_d(\mathbb{Z})$ and consider the random walk defined by $ρ$ on the torus $\mathbb{T}^d = \mathbb{R}^d/\mathbb{Z}^d$. Bourgain, Furmann, Lindenstrauss and Mozes proved that under an assumption on the group generated by the support of $$ρ$$, the random walk starting at any irrational point equidistributes in the torus. In this article, we study the central limit theorem and the law of the iterated logarithm for this walk starting at some point having good diophantine properties.
We show that one-dimensional random walks in a quasi-periodic environment with Liouville frequency generically have an erratic statistical behavior. In the recurrent case we show that neither quenched nor annealed limit theorems hold and both drift and variance exhibit wild oscillations, being logarithmic at some times and almost linear at other times. In the transient case we show that the annealed Central Limit Theorem fails generically. These results are in stark contrast with the Diophantine case where the Central Limit Theorem with linear drift and variance was established by Sinai.
We present a new method of proving the Diophantine extremality of various dynamically defined measures, vastly expanding the class of measures known to be extremal. This generalizes and improves the celebrated theorem of Kleinbock and Margulis ('98) resolving Sprindžuk's conjecture, as well as its extension by Kleinbock, Lindenstrauss, and Weiss ('04), hereafter abbreviated KLW. As applications we prove the extremality of all hyperbolic measures of smooth dynamical systems with sufficiently large Hausdorff dimension, and of the Patterson--Sullivan measures of all nonplanar geometrically finite…
