Maximal Hausdorff Dimension Research Articles

Self-similar sets and self-similar measures in the $p$-adics

In this paper, we investigate p -adic self-similar sets and p -adic self-similar measures. We introduce a condition (C) under which p -adic self-similar sets can be shown to have a number of nice properties. It is shown that p -adic self-similar sets satisfying condition (C) are p -adic path set fractals. This allows us to easily compute the Hausdorff dimension of these sets. We further show that the set of p -adic path set fractals is strictly larger than this set of p -adic self-similar sets. The directed graph associated to p -adic self-similar sets satisfying condition (C) is shown to have a unique essential class. Moreover, it is shown that almost all points are eventually in the essential class. For p -adic self-similar measures satisfying this condition, we show that many results involving local dimension are similar to those of their real counterparts, with fewer complications. We next study the more general p -adic path set fractals, first showing that the existence of an interior point is equivalent to the set having Hausdorff dimension 1 . We further show that often the decimation of p -adic path set fractals results in a set with maximal Hausdorff dimension.

A Peak Set with a Maximal Hausdorff Dimension on Each Slice

We consider a bounded balanced strictly convex domain Omega subset {mathbb {C}}^{d} with C^{2} boundary. Then there exists a peak set E with Hausdorff dimension equal to 1 on each slice. In particular E has maximal possible Hausdorff dimension equal to 2d-1.

Note on the Generalized Branching Random Walk on the Galton–Watson Tree

Let ∂T be a super-critical Galton–Watson tree. Recently, the first author computed almost surely and simultaneously the Hausdorff dimensions of the sets of infinite branches of the boundary of ∂T along which the sequence SnX(t)/SnX˜(t) has a given set of limit points, where SnX(t) and SnX˜(t) are two branching random walks defined on ∂T. In this study, we are interested in the study of the speed of convergence of this sequence. More precisely, for a given sequence s=(sn), we consider Eα,s=t∈∂T:SnX(t)−αSnX˜(t)∼snasn→+∞. We will give a sufficient condition on (sn) so that Eα,s has a maximal Hausdorff and packing dimension.

Topological flows for hyperbolic groups

AbstractWe show that for every non-elementary hyperbolic group the Bowen–Margulis current associated with a strongly hyperbolic metric forms a unique group-invariant Radon measure class of maximal Hausdorff dimension on the boundary square. Applications include a characterization of roughly similar hyperbolic metrics via mean distortion.

Simultaneous dense and nondense orbits for noncommuting toral endomorphisms

Let S and T be hyperbolic endomorphisms of \({\mathbb {T}}^d\) with the property that the span of the subspace contracted by S along with the subspace contracted by T is \({\mathbb {R}}^d\). We show that the intersection of the set of points with equidistributing orbit under S with the set of points with nondense orbit under T has maximal Hausdorff dimension. In the case that S and T are quasihyperbolic automorphisms, we prove that the Hausdorff dimension of the intersection is again maximal when we assume that \({\mathbb {R}}^d\) is spanned by the subspaces contracted by S and T along with the central eigenspaces of S and T.

Cantor-winning sets and their applications

We introduce and develop a class of Cantor-winning sets that share the same amenable properties as the classical winning sets associated to Schmidt's (α,β)-game: these include maximal Hausdorff dimension, invariance under countable intersections with other Cantor-winning sets and invariance under bi-Lipschitz homeomorphisms. It is then demonstrated that a wide variety of badly approximable sets appearing naturally in the theory of Diophantine approximation fit nicely into our broad-reaching framework. As applications of this phenomenon we answer several previously open questions, including some related to the Mixed Littlewood conjecture and approximation by multiplicative semigroups of integers.

Hausdorff and packing dimension of fibers and graphs of prevalent continuous maps

The notions of shyness and prevalence generalize the property of being zero and full Haar measure to arbitrary (not necessarily locally compact) Polish groups. The main goal of the paper is to answer the following question: What can we say about the Hausdorff and packing dimension of the fibers of prevalent continuous maps?Let K be an uncountable compact metric space. We prove that a prevalent f∈C(K,Rd) has many fibers with almost maximal Hausdorff dimension. This generalizes a theorem of Dougherty and yields that a prevalent f∈C(K,Rd) has graph of maximal Hausdorff dimension, generalizing a result of Bayart and Heurteaux. We obtain similar results for the packing dimension.We show that for a prevalent f∈C([0,1]m,Rd) the set of y∈f([0,1]m) for which dimH⁡f−1(y)=m contains a dense open set having full measure with respect to the occupation measure λm∘f−1, where dimH and λm denote the Hausdorff dimension and the m-dimensional Lebesgue measure, respectively. We also prove an analogous result when [0,1]m is replaced by any self-similar set satisfying the open set condition.We cannot replace the occupation measure with Lebesgue measure in the above statement: We show that the functions f∈C[0,1] for which positively many level sets are singletons form a non-shy set in C[0,1]. In order to do so, we generalize a theorem of Antunović, Burdzy, Peres and Ruscher. As a complementary result we prove that the functions f∈C[0,1] for which dimH⁡f−1(y)=1 for all y∈(min⁡f,max⁡f) form a non-shy set in C[0,1].We also prove sharper results in which large Hausdorff dimension is replaced by positive measure with respect to generalized Hausdorff measures, which answers a problem of Fraser and Hyde.

FINITELY CONSTRAINED GROUPS OF MAXIMAL HAUSDORFF DIMENSION

We prove that if $G_{P}$ is a finitely constrained group of binary rooted tree automorphisms (a group binary tree subshift of finite type) defined by an essential pattern group $P$ of pattern size $d$, $d\geq 2$, and if $G_{P}$ has maximal Hausdorff dimension (equal to $1-1/2^{d-1}$), then $G_{P}$ is not topologically finitely generated. We describe precisely all essential pattern groups $P$ that yield finitely constrained groups with maximal Hausdorff dimension. For a given size $d$, $d\geq 2$, there are exactly $2^{d-1}$ such pattern groups and they are all maximal in the group of automorphisms of the finite rooted regular tree of depth $d$.

Functions from Sobolev and Besov Spaces with Maximal Hausdorff Dimension of the Exceptional Lebesgue Set

We prove that for p > 1 and 0 < α < n/p there exists a function from the Bessel potentials class J α (L p (ℝ n )) such that the Hausdorff dimension of its exceptional Lebesgue set is n − αp. We also show that such a function may be taken from the Besov class B , (L p (ℝ n )) with any q > 0.

Boundary values of harmonic gradients and differentiability of Zygmund and Weierstrass functions

We study differentiability properties of Zygmund functions and series of Weierstrass type in higher dimensions. While such functions may be nowhere differentiable, we show that, under appropriate assumptions, the set of points where the incremental quotients are bounded has maximal Hausdorff dimension.

An Example of a Nearly Integrable Hamiltonian System with a Trajectory Dense in a Set of Maximal Hausdorff Dimension

The famous ergodic hypothesis suggests that for a typical Hamiltonian on a typical energy surface nearly all trajectories are dense. KAM theory disproves it. Ehrenfest (The Conceptual Foundations of the Statistical Approach in Mechanics. Ithaca, NY: Cornell University Press, 1959) and Birkhoff (Collected Math Papers. Vol 2, New York: Dover, pp 462–465, 1968) stated the quasi-ergodic hypothesis claiming that a typical Hamiltonian on a typical energy surface has a dense orbit. This question is wide open. Herman (Proceedings of the International Congress of Mathematicians, Vol II (Berlin, 1998). Doc Math 1998, Extra Vol II, Berlin: Int Math Union, pp 797–808, 1998) proposed to look for an example of a Hamiltonian near \({H_0(I)= \frac{\langle I, I \rangle}{2}}\) with a dense orbit on the unit energy surface. In this paper we construct a Hamiltonian \({H_0(I)+\varepsilon H_1(\theta , I , \varepsilon)}\) which has an orbit dense in a set of maximal Hausdorff dimension equal to 5 on the unit energy surface.

Nonadditive measure-theoretic pressure and applications to dimensions of an ergodic measure

AbstractWithout any additional conditions on subadditive potentials, this paper defines subadditive measure-theoretic pressure, and shows that the subadditive measure-theoretic pressure for ergodic measures can be described in terms of measure-theoretic entropy and a constant associated with the ergodic measure. Based on the definition of topological pressure on non-compact sets, we give another equivalent definition of subadditive measure-theoretic pressure, and obtain an inverse variational principle. This paper also studies the superadditive measure-theoretic pressure which has similar formalism to the subadditive measure-theoretic pressure. As an application of the main results, we prove that an average conformal repeller admits an ergodic measure of maximal Hausdorff dimension. Furthermore, for each ergodic measure supported on an average conformal repeller, we construct a set whose dimension is equal to the dimension of the measure.

Strong bifurcation loci of full Hausdorff dimension

In the moduli space $\mathcal{M}_d$ of degree $d$ rational maps, the bifurcation locus is the support of a closed $(1,1)$ positive current $T_\bif$ which is called the bifurcation current. This current gives rise to a measure $\mu_\bif:=(T_\bif)^{2d-2}$ whose support is the seat of strong bifurcations. Our main result says that $\supp(\mu_\bif)$ has maximal Hausdorff dimension $2(2d-2)$. As a consequence, the set of degree $d$ rational maps having $2d-2$ distinct neutral cycles is dense in a set of full Hausdorff dimension.

MAXIMAL DIMENSIONS OF UNIFORM SIERPINSKI FRACTALS

We studied the invariant fractals in convex N-gons given by N identical pure contractions at its vertices with non-overlapping images, which we called uniform Sierpinski fractals. We provided the explicit formulae for the maximal contraction ratio RN, and the maximal Hausdorff dimension hN, for the uniform Sierpinski fractals. We used maximal N-grams and principal crossing points as the main tools.

ON SHRINKING TARGETS FOR ℤ m ACTIONS ON TORI

Let A be an n×m matrix with real entries. Consider the set BadA of x∈[0,1)n for which there exists a constant c(x)>0 such that for any q∈ℤm the distance between x and the point {Aq} is at least c(x)|q|−m/n. It is shown that the intersection of BadA with any suitably regular fractal set is of maximal Hausdorff dimension. The linear form systems investigated in this paper are natural extensions of irrational rotations of the circle. Even in the latter one-dimensional case, the results obtained are new.

ON BADLY APPROXIMABLE COMPLEX NUMBERS

AbstractWe show that the set of complex numbers which are badly approximable by ratios of elements of the ring of integers in $\(\mathbb{Q}(\sqrt{-D})\)$, where D ∈ {1, 2, 3, 7, 11, 19, 43, 67, 163} has maximal Hausdorff dimension. In addition, the intersection of these sets is shown to have maximal dimension. The results remain true when the sets in question are intersected with a suitably regular fractal set.

Natural invariant measures, divergence points and dimension in one-dimensional holomorphic dynamics

AbstractIn this paper we discuss the dimension-theoretical properties of rational maps on the Riemann sphere. In particular, we study the existence and uniqueness of generalized physical measures for several classes of maps including hyperbolic, parabolic, non-recurrent and topological Collet–Eckmann maps. These measures have the property that their typical points have maximal Hausdorff dimension. On the other hand, we prove that the set of divergence points (the set of points which are non-typical for any invariant measure) also has maximal Hausdorff dimension. Finally, we prove that if (fa)ais a holomorphic family of stable rational maps, then the dimensiond(fa) is a continuous and plurisubharmonic function of the parametera. In particular,d(f) varies continuously and plurisubharmonically on an open and dense subset ofRatd, the space of all rational maps with degreed≥2.

On the definition of Bourgain points

This paper is devoted to study of the so-called Bourgain points (B-points) for functions from L∞(ℝ). In 1993, Bourgain showed that for a real-valued bounded function f, the set Ef of B-points is everywhere dense and has the maximal Hausdorff dimension, dim H(Ef ) = 1; in addition, the vertical variation of the harmonic extension of f to the upper half-plane is finite at B-points. An essentially simpler definition of B-points is given compared to that in the original works by Bourgain. A geometric characterization of B-points for Cantor-like sets is obtained. Bibliography: 7 titles.

Measures of Maximal Dimension for Linear Horseshoes

We consider a linear Smale-William's' horseshoe with different contraction/dilatation coefficients and find equilibrium states of maximal Hausdorff dimension. We compute this dimension and show an example when the state of maximal dimension is non-unique.

New phenomena associated with homoclinic tangencies

We survey some recently obtained generic consequences of the existence of homoclinic tangencies in diffeomorphisms of surfaces. Among other things, it has been shown that they give rise to invariant topologically transitive sets with maximal Hausdorff dimension, that they prohibit the existence of various kinds of symbolic extensions and that they form an impediment to the existence of Sinai–Ruelle–Bowen (SRB) measures. The main new result described here, together with a positive answer to an as yet unproved conjecture of Palis, would prove that generically on surfaces, SRB measures only exist on uniformly hyperbolic attractors.