Hausdorff Dimension Estimates Research Articles

Dimension Analysis via Differential Lyapunov and Dissipation Inequalities⋆

Motivated by complex and diverse dynamics in engineering and nature, this paper considers Hausdorff dimension estimates for compact invariant sets of dynamical systems. Our aim is to develop a framework of dimension analysis as an extension of Lyapunov's stability theory as well as Willems’ dissipativity theory. In particular, we extend the existing result by Smith involving Lyapunov inequalities for Hausdorff dimension estimates by employing state-dependent metrics. The obtained result is then combined with the dissipativity framework for studying interconnected systems. This paper is mainly devoted to mathematics but describes several connections with recent works on the differential framework for contraction analysis.

Hausdorff dimension estimates applied to Lagrange and Markov spectra, Zaremba theory, and limit sets of Fuchsian groups

In this note we will describe a simple and practical approach to get rigorous bounds on the Hausdorff dimension of limits sets for some one dimensional Markov iterated function schemes. The general problem has attracted considerable attention, but we are particularly concerned with the role of the value of the Hausdorff dimension in solving conjectures and problems in other areas of mathematics. As our first application we confirm, and often strengthen, conjectures on the difference of the Lagrange and Markov spectra in Diophantine analysis, which appear in the work of Matheus and Moreira [Comment. Math. Helv. 95 (2020), pp. 593–633]. As a second application we (re-)validate and improve estimates connected with the Zaremba conjecture in number theory, used in the work of Bourgain–Kontorovich [Ann. of Math. (2) 180 (2014), pp. 137–196], Huang [An improvement to Zaremba’s conjecture, ProQuest LLC, Ann Arbor, MI, 2015] and Kan [Mat. Sb. 210 (2019), pp. 75–130]. As a third more geometric application, we rigorously bound the bottom of the spectrum of the Laplacian for infinite area surfaces, as illustrated by an example studied by McMullen [Amer. J. Math. 120 (1998), pp. 691-721]. In all approaches to estimating the dimension of limit sets there are questions about the efficiency of the algorithm, the computational effort required and the rigour of the bounds. The approach we use has the virtues of being simple and efficient and we present it in this paper in a way that is straightforward to implement. These estimates apparently cannot be obtained by other known methods.

On exceptional sets in the metric Poissonian pair correlations problem

Let left( a_{n}right) _{n} be a strictly increasing sequence of positive integers. Recent works uncovered a close connection between the additive energy Eleft( A_{N}right) of the cut-offs A_{N}=left{ a_{n},{:},,nle Nright} , and left( a_{n}right) _{n} possessing metric Poissonian pair correlations which is a metric version of a uniform distribution property of “second order”. Firstly, the present article makes progress on a conjecture of Aichinger, Aistleitner, and Larcher; by sharpening a theorem of Bourgain which states that the set of alpha in left[ 0,1right] satisfying that left( leftlangle alpha a_{n}rightrangle right) _{n} with Eleft( A_{N}right) =Omega left( N^{3}right) does not have Poissonian pair correlations has positive Lebesgue measure. Secondly, we construct sequences with high additive energy which do not have metric Poissonian pair correlations, in a strong sense, and provide Hausdorff dimension estimates.

Structure of sets which are well approximated by zero sets of harmonic polynomials

The zero sets of harmonic polynomials play a crucial role in the study of the free boundary regularity problem for harmonic measure. In order to understand the fine structure of these free boundaries a detailed study of the singular points of these zero sets is required. In this paper we study how "degree $k$ points" sit inside zero sets of harmonic polynomials in $\mathbb R^n$ of degree $d$ (for all $n\geq 2$ and $1\leq k\leq d$) and inside sets that admit arbitrarily good local approximations by zero sets of harmonic polynomials. We obtain a general structure theorem for the latter type of sets, including sharp Hausdorff and Minkowski dimension estimates on the singular set of "degree $k$ points" ($k\geq 2$) without proving uniqueness of blowups or aid of PDE methods such as monotonicity formulas. In addition, we show that in the presence of a certain topological separation condition, the sharp dimension estimates improve and depend on the parity of $k$. An application is given to the two-phase free boundary regularity problem for harmonic measure below the continuous threshold introduced by Kenig and Toro.

Improved Hausdorff dimension estimate for vertical projections in the Heisenberg group

We explore the effect of vertical projections on the Hausdorff dimension of sets in the Heisenberg group endowed with the Koranyi distance. It is known that the dimension of an at most one-dimensional set generically does not decrease under such mappings. The proof uses a potential-theoretic approach which, for higher dimensional sets, only yields a trivial lower bound. In the present note, we provide an improved estimate for the dimension and thus prove that the previous trivial bound is not sharp. Moreover, for the larger family of projections onto cosets of vertical subgroups, we show that the potential-theoretic approach can be applied to establish almost sure dimension conservation for sets of dimension up to two.

Regularity and capacity for the fractional dissipative operator

This paper is devoted to exploring some analytic–geometric properties of the regularity and capacity associated with the so-called fractional dissipative operator ∂t+(−Δ)α, naturally establishing a diagonally sharp Hausdorff dimension estimate for the blow-up set of a weak solution to the fractional dissipative equation (∂t+(−Δ)α)u(t,x)=F(t,x) subject to u(0,x)=0. The methods used in this paper rely on effectively controlling the time-dependent non-local kernels and potentials with fractional order α∈(0,1), dual representation of the capacity and Frostman type theorem from geometric measure theory.

Hausdorff dimension estimates for restricted families of projections in [formula omitted

This paper is concerned with restricted families of projections in R3. Let K⊂R3 be a Borel set with Hausdorff dimension dim⁡K=s>1. If G is a smooth and sufficiently well-curved one-dimensional family of two-dimensional subspaces, the main result states that there exists σ(s)>1 such that dim⁡πV(K)≥σ(s) for almost all V∈G. A similar result is obtained for some specific families of one-dimensional subspaces.

Bifurcations of invariant measures in discrete-time parameter dependent cocycles

We consider parameter-dependent cocycles generated by nonautonomous difference equations. One of them is a discrete-time cardiac conduction model. For this system with a control variable a cocycle formulation is presented. We state a theorem about upper Hausdorff dimension estimates for cocycle attractors which includes some regulating function. We also consider the existence of invariant measures for cocycle systems using some elements of Perron-Frobenius theory and discuss the bifurcation of parameter-dependent measures.

Boundary blow-up under Sobolev mappings

We prove that for mappings $W^{1,n}(B^n, \R^n),$ continuous up to the boundary, with modulus of continuity satisfying certain divergence condition, the image of the boundary of the unit ball has zero $n$-Hausdorff measure. For H\"older continuous mappings we also prove an essentially sharp generalized Hausdorff dimension estimate.

A note on zero-sets of fractional sobolev functions with negative power of integrability

We extend a Poincare-type inequality for functions with large zero-sets by Jiang and Lin to fractional Sobolev spaces. As a consequence, we obtain a Hausdorff dimension estimate on the size of zero sets for fractional Sobolev functions whose inverse is integrable. Also, for a suboptimal Hausdorff dimension estimate, we give a completely elementary proof based on a pointwise Poincare-style inequality.

WITHDRAWN: Hausdorff dimension estimate of compact invariant sets for stochastic dynamical systems

This article has been withdrawn at the request of the author(s) and/or editor. The Publisher apologizes for any inconvenience this may cause. The full Elsevier Policy on Article Withdrawal can be found at http://www.elsevier.com/locate/withdrawalpolicy .

Boundary behavior of conformal deformations

We study conformal deformations of the Euclidean metric in the unit ball B n \mathbb {B}^{n} . We assume that the density associated with the deformation satisfies a Harnack inequality and an arbitrary volume growth condition on the isodiametric profile. We establish a Hausdorff (gauge) dimension estimate for the set E ⊂ ∂ B n E\subset \partial \mathbb {B}^{n} where such a deformation mapping can “blow up”. We also prove a generalization of Gerasch’s theorem in our setting.

Fractal dimension estimate for invariant set in complete Riemannian manifold

Abstract In this paper, we give a simple upper bounds for the fractal dimensions of a forward invariant set and a negatively invariant set of a C1-diffeomorphism of the complete Riemannian manifold with non-negative Ricci curvature. These results generalize the corresponding result of C. Robinson [Dynamics systems: stability symbolic dynamics and chaos. 2nd ed. Boca Raton, London, New York, Washington: CRC Press; 1999] in R n , and weakens the condition to the singular values of the tangent map in [Boichenko VA, Franz A, Leonov GA, Reitmann V. Hausdorff and fractal dimension estimates for invariant sets of non-injective maps. Z Anal Anw 1998;17:207–23]. Finally, as application, we give the upper bound of the fractal dimension of an invariant set of the flow on Riemannian manifold.

Transport density in Monge-Kantorovich problems with Dirichlet conditions

We study the properties of the transport density measure in the Monge-Kantorovich optimal mass transport problem in the presence of so-called Dirichlet constraint, i.e. when some closed set is given along which the cost of transportation is zero. The Hausdorff dimension estimates, as well as summability and higher regularity properties of the transport density are studied. The uniqueness of the transport density is proven in the case when the masses to be transported are represented by measures absolutely continuous with respect to the Lebesgue measure.

Upper and lower Hausdorff dimension estimates for invariant sets of weak k‐1‐endomorphisms

AbstractFor a special class of non‐injective maps called the weak k‐1‐endomorphisms on Riemannian manifolds upper and lower bounds for Hausdorff dimensions of invariant sets are given in terms of the singular values of the tangent maps, which generalize Franz's corresponding results. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

ZERO SET OF SOBOLEV FUNCTIONS WITH NEGATIVE POWER OF INTEGRABILITY

Here the authors are interested in the zero set of Sobolev functions and functions of bounded variation with negative power of integrability. The main result is a general Hausdorff dimension estimate on the size of zero set. The research is motivated by the model on van der waal force driven thin film, which is a singular elliptic equation. After obtaining some basic regularity result, the authors get an estimate on the size of singular set; such set corresponds to the thin film rupture set in the thin film model.

An obstacle-problem-like equation with two phases: pointwise regularity of the solution and an estimate of the Hausdorff dimension of the free boundary

where λ+ 0 and λ+ + λ− > 0. Equation (1) is related to the time-dependent equation 0 = α∂t max(v, 0)+ β∂t min(v, 0)−∆v in (0, T ) × Ω which has been used to describe an instantaneous and complete reaction of two substances coming into contact at a surface Γ (see [2, 3] and [6]). The difficulty one confronts in this Stefan-like problem is that the interface {v = 0} consists in general of two parts—one where the gradient of v is nonzero and one where the gradient of v vanishes. At the latter part we expect the gradient of v to have linear growth in space. However, because of the decomposition into two different types of growth, it is not possible to derive a growth estimate by, for example, a Bernstein technique. Assuming that α > β > 0 and that the time derivative ∂tv is non-negative and Holder continuous near some free boundary point (t0, x0), v(t0) is a solution of (1) with Holder continuous coefficients λ+ and λ−. To that our result applies and yields the expected C1,1-regularity of v(t0) in a pointwise sense and, for positive time derivative, the Hausdorff dimension estimate dim(∂{v(t0) > 0} ∪ ∂{v(t0) < 0}) n − 1 (see Proposition 4.1, Remark 4.1, Corollary 5.1 and Remark 5.1).

Self-Similar Blow-Up and Hausdorff Dimension Estimates for a Class of Parabolic Free Boundary Problems

We consider variational solutions of the quenching problem $\partial_t u - \Delta u = - u^\gamma \chi_{\{u>0\}}$ with exponent $\gamma \in (-{1\over 3},0)$ and solutions of the heat equation with Bernoulli-type condition on the free boundary which arises in combustion theory, $\partial_t v - \Delta v =0$ in $\{v>0\}\>,$ $\vert \nabla v \vert = 1$ on $\partial\{v>0\}.$ We show that blow-up limits of u and v are backward self-similar solutions and use this to determine the Hausdorff dimension of the free boundaries $\partial\{u>0\}$ and $\partial\{v>0\}.$

Two boundary value problems for the Ginzburg-Landau equation

Two boundary value problems for the Ginzburg-Landau equation are considered. Extensive numerical calculations have been performed in each case, including bifurcation histories, spectral analysis, Poincaré sections and Hausdorff dimension estimates. The approach to the inviscid limit is given detailed treatment. In this case universal behavior has been found to exist. Arguments are presented to account for this behavior.