J. Gómez, H. Pinedo, C. Uzcátegui
We provide a sufficient condition for a topological partial action of a Hausdorff group on a metric space is continuous, provide that it is separately continuous.
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The Hausdorff dimension of general Sierpinski carpets, [4] and [20], and the generalization on Lalley-Gatzouras carpets, [10], are today well known results, the formulas being obtain via the variational principle for the dimension. We call the multidimensional versions of these carpets Sierpinski sponges and self-affine sponges, respectively,. In this paper we show that the Hausdorff dimension of self-affine sponges, defined in R3, is a Lipschitz continuous function at Sierpinski sponges.
In a recent paper [Adv. Math. 305:165--196, 2017], Komornik et al.~proved a long-conjectured formula for the Hausdorff dimension of the set $\mathcal{U}_q$ of numbers having a unique expansion in the (non-integer) base $q$, and showed that this Hausdorff dimension is continuous in $q$. Unfortunately, their proof contained a gap which appears difficult to fix. This article gives a completely different proof of these results, using a more direct combinatorial approach.
Continuity of Hausdorff Dimension at Hopf Bifurcation
研究通过Hopf分岔从传递Anosov微分同胚得到的非双曲排斥子的Hausdorff维数和盒维数在参数分岔处的连续性，通过带孔映射方法证明。
We show continuity under equivariant Gromov-Hausdorff convergence of the critical exponent of discrete, non-elementary, torsion-free, quasiconvex-cocompact groups with uniformly bounded codiameter acting on uniformly Gromov-hyperbolic metric spaces.
Let $φ_0$ be a smooth area-preserving diffeomorphism of a compact surface $M$ and let $Λ_0$ be a horseshoe of $φ_0$ with Hausdorff dimension strictly smaller than one. Given a smooth function $f:M\to \mathbb{R}$ and a small smooth area-preserving perturtabion $φ$ of $φ_0$, let $L_{φ, f}$, resp. $M_{φ, f}$ be the Lagrange, resp. Markov spectrum of asymptotic highest, resp. highest values of $f$ along the $φ$-orbits of points in the horseshoe $Λ$ obtained by hyperbolic continuation of $Λ_0$. We show that, for generic choices of $φ$ and $f$, the Hausdorff dimension of the sets $L_{φ, f}\cap (-\in…
We study continuity and discontinuity properties of some popular measure-dimension mappings under some topologies on the space of probability measures in this work. We give examples to show that no continuity can be guaranteed under general weak, setwise or TV topology on the space of measures for any of these measure-dimension mappings. However, in some particular circumstances or by assuming some restrictions on the measures, we do have some (semi-)continuity results. We then apply our continuity results to concerning measures appearing in two kinds of infinite iterated function systems, nam…
Given a non-conformal repeller $Λ$ of a $C^{1+γ}$ map, we study the Hausdorff dimension of the repeller and continuity of the sub-additive topological pressure for the sub-additive singular valued potentials. Such a potential always possesses an equilibrium state. We then use a substantially modified version of Katok's approximating argument, to construct a compact invariant set on which the corresponding dynamical quantities (such as Lyapunov exponents and metric entropy) are close to that of the equilibrium measure. This allows us to establish continuity of the sub-additive topological press…
Let $g_0$ be a smooth pinched negatively curved Riemannian metric on a complete surface $N$, and let $Λ_0$ be a basic hyperbolic set of the geodesic flow of $g_0$ with Hausdorff dimension strictly smaller than two. Given a small smooth perturbation $g$ of $g_0$ and a smooth real-valued function $f$ on the unit tangent bundle to $N$ with respect to $g$, let $L_{g,Λ,f}$, resp. $M_{g,Λ,f}$ be the Lagrange, resp. Markov spectrum of asymptotic highest, resp. highest values of $f$ along the geodesics in the hyperbolic continuation $Λ$ of $Λ_0$. We prove that, for generic choices of $g$ and $f$, the …
Amenable Metric Mean Dimension and Amenable Mean Hausdorff Dimension of Product Sets and Metric Varying
本文研究了可顺应群作用下度量平均维数和平均Hausdorff维数关于度量的连续性，推广了已有结果，并证明了这两种维数的乘积公式。
Consider a parameter dependent vector field on either Euclidean space or a compact Riemannian manifold. Suppose that it possesses a parameter dependent initial condition and a parameter dependent stable hyperbolic equilibrium point. It is valuable to determine the set of parameter values, which we call the recovery set, whose corresponding initial conditions lie within the region of attraction of the corresponding stable equilibrium point. A boundary parameter value is a parameter value whose corresponding initial condition lies in the boundary of the region of attraction of the corresponding …
On the conditions for the continuity of the Hausdorff measure
本文研究了迭代函数系统中Hausdorff测度连续性的条件，证明了在特定条件下Hausdorff测度收敛于1，并给出了非线性函数IFS的四个保证条件及示例。
Continuity of fractal dimensions in conservative generic Markov and Lagrange dynamical spectra
本文研究了保守泛动力学谱中分形维度的连续性，证明了在泛泛选择下，拉格朗日谱和马尔可夫谱的豪斯多夫维数相等，并推广了Cerqueira-Matheus-Moreira定理。
We establish sufficient conditions for the upper semicontinuity and the continuity of the entropy of Sinai probability measures invariant by partially hyperbolic diffeomorphisms and discuss their application in several examples.
Topological entropy and Hausdorff dimension of shrinking target sets
本文针对具有指数 specification 性质和 Lipschitz 连续性的动力系统，建立收缩目标集拓扑熵与 Hausdorff 维数的上下界，并在环面双曲、扩张及若干符号系统中证明二者重合。
In this paper, we consider certain partially hyperbolic diffeomorphisms with center of arbitrary dimension and obtain continuity properties of the topological entropy under $C^1$ perturbations. The systems considered have subexponential growth in the center direction and uniform exponential growth along the unstable foliation. Our result applies to partially hyperbolic diffeomorphisms which are Lyapunov stable in the center direction. It applies to another important class of systems which do have subexponential growth in the center direction, for which we develop a technique to use exponential…
Continuity properties of partial entropy
本文研究了$C^{1+α}$微分同胚中偏熵的上半连续性，证明了当Lyapunov指数之和连续时该性质成立，扩展了Newhouse关于光滑微分同胚的结果。
We prove that complex Bernoulli convolutions are absolutely continuous in the supercritical parameter region, outside of an exceptional set of parameters of zero Hausdorff dimension. Similar results are also obtained in the biased case, and for other parametrized families of self-similar sets and measures in the complex plane, extending earlier results.
We systematically investigate examples of non-hyperbolic dynamical systems having irregular sets of full topological entropy and full Hausdorff dimension. The examples include some partially hyperbolic systems and geometric Lorenz flows. We also pose interesting questions for future research.
In this manuscript the concept of hyperspace is revisited. The main purpose is to study hyperconvergence and continuity of orbital and limit set functions for semigroup action on completely regular space. Some general facts on Hausdorff and Kuratowski hyperconvergence are presented.
Hausdorff dimension of sets of continued fractions with bounded odd and even order partial quotients
研究奇数阶部分商为1的连分数集合的大小，证明两个此类集合的和与积均包含非空区间，并确定奇数阶部分商为1且偶数阶部分商以指数、超指数及一般正函数速率增长时集合的Hausdorff维数。
The existence and construction of periodic approximations with convergent spectra is crucial in solid state physics for the spectral study of corresponding Schrödinger operators. In a forthcoming work [9] (arXiv:1709.00975) this task was boiled down to the existence and construction of periodic approximations of the underlying dynamical systems in the Hausdorff topology. As a result the one-dimensional systems admitting such approximations are completely classified in the present work. In addition explicit constructions are provided for dynamical systems defined by primitive substitutions cove…
I. J. Good (1941) showed that the set of irrational numbers in $(0,1)$ whose partial quotients $a_n$ tend to infinity is of Hausdorff dimension $1/2$. A number of related results impose restrictions of the type $a_n\in B$ or $a_n\geq f(n)$, where $B$ is an infinite subset of $\mathbb N$ and $f$ is a rapidly growing function with $n$. We show that, for an arbitrary $B$ and an arbitrary $f$ with values in $[\min B,\infty)$ and tending to infinity, the set of irrational numbers in $(0,1)$ such that \[ a_n\in B,\ a_n\leq f(n)\text{ for all $n\in\mathbb N$, and }a_n\to\infty\text{ as }n\to\infty\] …
We provide conditions which imply the continuity of the Lyapunov exponents for non-uniformly fiber-bunched cocycles in $SL(2,\mathbb{R})$. The main theorem is an extension of the result of Backes, Brown and Butler and gives a partial answer to a conjecture of Marcelo Viana.
Hausdorff dimension of sets with restricted, slowly growing partial quotients in semi-regular continued fractions
本研究确定了半正则连分数中部分商受限且缓慢增长的无理数集合的豪斯多夫维数，推广了正则连分数的相关结果，通过构造非自治迭代函数系统并应用Bowen公式证明。
We study the spectral location of strongly pattern equivariant Hamiltonians arising through configurations on a colored lattice. Roughly speaking, two configurations are "close to each other" if, up to a translation, they "almost coincide" on a large fixed ball. The larger this ball is, the more similar they are, and this induces a metric on the space of the corresponding dynamical systems. Our main result states that the map which sends a given configuration into the spectrum of its associated Hamiltonian, is Hölder (even Lipschitz) continuous in the usual Hausdorff metric. Specifically, the …
Continuity of the Julia sets in non-archimedean and hybrid settings
该研究针对完备非阿基米德域上Berkovich空间及带混合范数的复空间参数化的单变量有理函数族，证明了朱利亚集在豪斯多夫拓扑下连续性的类似Mané--Sad--Sullivan定理结果。
Hausdorff dimension of sets of continued fractions with unbounded partial quotients along subsequence
研究沿给定子序列的部分商趋于无穷的连分数集合的Hausdorff维数，证明维数由子序列的上密度决定：上密度为正时维数为1/2，上密度为零时维数为1。
Hausdorff dimension of sets of numbers whose continued fractions contain arbitrarily long arithmetic progressions
本文研究了偏商序列严格递增且包含任意长等差数列的无理数的豪斯多夫维数。
We present a detailed Hausdorff dimension analysis of the set of real numbers where the product of consecutive partial quotients in their continued fraction expansion grow at a certain rate but the growth of the single partial quotient is at a different rate. We consider the set \begin{equation*} \FF(Φ_1,Φ_2) \defeq \EE(Φ_1) \backslash \EE(Φ_2)=\left\{x\in[0,1): \begin{split} a_n(x)a_{n+1}(x) & \geqΦ_1(n) \text{\,\, for infinitely many } n\in\N a_{n+1}(x) & <Φ_2(n) \text{\,\, for all sufficiently large } n\in\N \end{split} \right\}, \end{equation*} where $Φ_i:\N\to(0,\infty)$ are any functions…
Let $α$ be an irrational real number. We show that the set of $ε$-badly approximable numbers \[ \mathrm{Bad}^\varepsilon (α) := \{x\in [0,1]\, : \, \liminf_{|q| \to \infty} |q| \cdot \| qα-x \| \geq \varepsilon \} \] has full Hausdorff dimension for some positive $ε$ if and only if $α$ is singular on average. The condition is equivalent to the average $\frac{1}{k} \sum_{i=1, \cdots, k} \log a_i$ of the logarithms of the partial quotients $a_i$ of $α$ going to infinity with $k$. We also consider one-sided approximation, obtain a stronger result when $a_i$ tends to infinity, and establish a part…
On the Hausdorff dimension and attracting laminations for fully irreducible automorphisms of free groups
研究自由群自同构相关抽象代数层叠端点集的豪斯多夫维数，对指数增长及环面完全不可约外自同构，其吸引层叠或终结层叠端点集的豪斯多夫维和填充维数为\(0\)，上盒维数\(>0\)。
Hölder continuity and laminarity of the Green currents for Hénon-like maps
研究Hénon-like映射的Green流的Hölder连续超势及层性，证明其最大熵测度为Monge-Ampère流，并具有正Hausdorff维数。
Hausdorff dimension of the set of singular pairs
本文证明了奇异对集的豪斯多夫维数为4/3。我们还证明了diag（e^t，e^t，e^{-2t}）在SL（3，R）/SL（3，Z）上的作用允许发散轨道以任意慢的预定速率退出到无穷远，回答了A.N.斯塔科夫作为分析的副产品，我们得到了连分数收敛所满足的基本不等式的高维推广。作为计算豪斯多夫维数的技术的一个例子，我们证明了具有发散的部分同分向量的实数集具有豪斯多夫维数1/2。
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