In this paper we study the dimension of a family of sets arising in open dynamics. We use exponential mixing results for diagonalizable flows in compact homogeneous spaces $X$ to show that the Hausdorff dimension of set of points that lie on trajectories missing a particular open ball of radius $r$ is at most $$\dim X + C\frac{r^{\dim X}}{\log r},$$ where $C>0$ is a constant independent of $r>0$. Meanwhile, we also describe a general method for computing the least cardinality of open covers of dynamical sets using volume estimates.
For a surface diffeomorphism, a compact invariant locally maximal set $W$ and some subset $A\subset W$ we study the $A$-exceptional set, that is, the set of points whose orbits do not accumulate at $A$. We show that if the Hausdorff dimension of $A$ is smaller than the Hausdorff dimension $d$ of some ergodic hyperbolic measure, then the topological entropy of the exceptional set is at least the entropy of this measure and its Hausdorff dimension is at least $d$. Particular consequences occur when there is some a priori defined hyperbolic structure on $W$ and, for example, if there exists an SR…
Let $I=[0,1)$, $-1<λ<1$ and $f\colon I\to I$ be a piecewise $λ$-affine map of the interval $I$, i.e., there exist a partition $0=a_0<a_1<\cdots< a_{k-1}<a_k=1$ of the interval $I$ into $k\geq2$ subintervals and $b_1,\ldots, b_k\in\mathbb{R}$ such that $f(x)=λx+ b_i$ for every $x\in[a_{i-1},a_{i})$ and $i=1,\ldots,k$. The exceptional set $\mathcal{E}_f$ of $f$ is the set of parameters $δ\in\mathbb{R}$ such that $R_δ\circ f$ is not asymptotically periodic, where $R_δ\colon I\to I$ is the rotation of angle $δ$. In this paper we prove that $\mathcal{E}_f$ has zero Hausdorff dimension. We derive th…
Given a rational map of the Riemann sphere and a subset $A$ of its Julia set, we study the $A$-exceptional set, that is, the set of points whose orbit does not accumulate at $A$. We prove that if the topological entropy of $A$ is less than the topological entropy of the full system then the $A$-exceptional set has full topological entropy. Furthermore, if the Hausdorff dimension of $A$ is smaller than the dynamical dimension of the system then the Hausdorff dimension of the $A$-exceptional set is larger than or equal to the dynamical dimension, with equality in the particular case when the dyn…
For one parameter subgroup action on a finite volume homogeneous space, we consider the set of points admitting divergent on average trajectories. We show that the Hausdorff dimension of this set is strictly less than the manifold dimension of the homogeneous space. As a corollary we know that the Hausdorff dimension of the set of points admitting divergent trajectories is not full, which proves a conjecture of Y. Cheung.
We consider (not self-similar) Cantor sets defined by a sequence of piecewise linear functions. We prove that the dimension of the harmonic measure on such a set is strictly smaller than its Hausdorff dimension. Some Hausdorff measure estimates for these sets are also provided.
Let $f: M \to M$ be a $C^{1+α}$ map/diffeomorphism of a compact Riemannian manifold $M$ and $μ$ be an expanding/hyperbolic ergodic $f$-invariant Borel probability measure on $M$. Assume $f$ is average conformal expanding/hyperbolic on the support set $W$ of $μ$ and $W$ is locally maximal. For any subset $A\subset W$ with small entropy or dimension, we investigate the topological entropy and Hausdorff dimensions of the $A$-exceptional set and the limit $A$-exceptional set.
In this paper we compute the Hausdorff dimension of the set D_n of points on divergent trajectories of the homogeneous flow induced by a certain one-parameter subgroup of G=SL(2,R) acting by left multiplication on the product space G^n/Gamma^n, where Gamma=SL(2,Z). We prove that the Hausdorff dimension of D_n equals 3n-(1/2) for any n greater than one.
It is known that hyperbolic dynamical systems admit a unique invariant probability measure with maximal entropy. We prove an effective version of this statement and use it to estimate an upper bound for Hausdorff dimension of exceptional sets arising from dynamics.
Mean Hausdorff dimension is a dynamical version of Hausdorff dimension. It provides a way to dynamicalize geometric measure theory. We pick up the following three classical results of fractal geometry. (1) The calculation of Hausdorff dimension of homogeneous sets in the circle. (2) The coincidence of Hausdorff and Minkowski dimensions for self-similar sets. (3) The calculation of Hausdorff dimension of Bedford--McMullen carpets. We develop their analogues for mean Hausdorff dimension: (1) The calculation of mean Hausdorff dimension of homogeneous systems in the infinite dimensional torus. (2)…
Recently, Einsiedler and the authors provided a bound in terms of escape of mass for the amount by which upper-semicontinuity for metric entropy fails for diagonal flows on homogeneous spaces $Γ\backslash G$, where $G$ is any connected semisimple Lie group of real rank 1 with finite center and $Γ$ is any nonuniform lattice in $G$. We show that this bound is sharp and apply the methods used to establish bounds for the Hausdorff dimension of the set of points which diverge on average.
We consider Bourgain's ergodic theorem regarding arithmetic averages in the cases where quantitative mixing is present in the dynamical system. Focusing on the case of the horocyclic flow, those estimates allows us to bound from above the Hausdorff dimension of the exceptional set, providing evidence towards conjectures by Margulis,Shah and Sarnak regarding equidistribution of arithmetic averages in homogeneous spaces. We also prove the existence of a uniform upper bound for the Hausdorff dimension of the exceptional set which is independent from the spectral gap.
In this paper, we give a definition of Diophantine points of type $γ$ for $γ\geq0$ in a homogeneous space $G/Γ$, and compute the Hausdorff dimension of the subset of points which are not Diophantine of type $γ$ when $G$ is a semisimple Lie group of real rank one. We also deduce a Jarník-Besicovitch Theorem on Diophantine approximation in Heisenberg groups.
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 functio…
The Ricci flow was introduced by Hamilton and gained its importance through the years. Of special importance is the limiting behavior of the flow and its symmetry properties. Taking this into account, we present a novel normalization for the homogeneous Ricci flow with natural compactness properties. In addition, we present a characterization for Gromov-Hausdorff limits of homogeneous spaces. As a result, we present a detailed picture of the homogeneous Ricci flow for three-isotropy-summands flag manifolds: phase portraits, basins of attractions, conjugation classes and collapsing phenomena. M…
We prove that the set of exceptional $λ\in (1/2,1)$ such that the associated Bernoulli convolution is singular has zero Hausdorff dimension, and likewise for biased Bernoulli convolutions, with the exceptional set independent of the bias. This improves previous results by Erdös, Kahane, Solomyak, Peres and Schlag, and Hochman. A theorem of this kind is also obtained for convolutions of homogeneous self-similar measures. The proofs are very short, and rely on old and new results on the dimensions of self-similar measures and their convolutions, and the decay of their Fourier transform.
This paper is concerned with the polynomial integrability of the two-dimensional Hamiltonian systems associated to complex homogeneous polynomial potentials of degree $k$ of type $V_{k,l}=α(q_2-i q_1)^l (q_2+iq_1)^{k-l}$ with $α\in\mathbb{C}$ and $l=0,1,\dots, k$, called exceptional potentials. Hietarinta \cite{Hietarinta1983} proved that the potentials with $l=0,1,k-1,k$ and $l=k/2$ for $k$ even are polynomial integrable. We present an elementary proof of this fact in the context of the polynomial bi-homogeneous potentials, as was introduced by Combot et al. \cite{Combot2020}. In addition, we…
We show that for any two homogeneous Cantor sets with sum of Hausdorff dimensions that exceeds 1, one can create an interval in the sumset by applying arbitrary small perturbations (without leaving the class of homogeneous Cantor sets). In our setting the perturbations have more freedom than in the setting of the Palis' conjecture, so our result can be viewed as an affirmative answer to a weaker form of the Palis' conjecture. We also consider self-similar sets with overlaps on the real line (not necessarily homogeneous), and show that one can create an interval by applying arbitrary small pert…
We show that, for any $0<γ<1/2$, any $(α,β)\in\mathbb{R}^2$ except on a set with Hausdorff dimension about $\sqrtγ$, any small $0<\varepsilon<1$ and any large $N\in\mathbb{N}$, the number of integers $n\in[1,N]$ such that $n\langle nα\rangle\langle nβ\rangle<\varepsilon$ is greater than $γ\varepsilon\log N$ up to a uniform constant. This can be seen as a quantitative result on the fact that the exceptional set to Littlewood's conjecture has Hausdorff dimension zero, obtained by M. Einsiedler, A. Katok and E. Lindenstrauss in 2000's. For the proof, we study the behavior of the empirical measure…
Let $X = G/Γ$, where $G$ is a Lie group and $Γ$ is a lattice in $G$, and let $U$ be a subset of $X$ whose complement is compact. We use the exponential mixing results for diagonalizable flows on $X$ to give upper estimates for the Hausdorff dimension of the set of points whose trajectories miss $U$. This extends a recent result of Kadyrov and produces new applications to Diophantine approximation, such as an upper bound for the Hausdorff dimension of the set of weighted uniformly badly approximable systems of linear forms, generalizing an estimate due to Broderick and Kleinbock.
Let $β> 1$ and the run-length function $r_n(x,β)$ be the maximal length of consecutive zeros amongst the first n digits in the $β$-expansion of $x\in[0,1]$. The exceptional set $$E_{\max}^φ=\left\{x \in [0,1]:\liminf_{n\rightarrow \infty}\frac{r_n(x,β)}{φ(n)}=0, \limsup_{n\rightarrow \infty}\frac{r_n(x,β)}{φ(n)}=+\infty\right\}$$ is investigated, where $φ: \mathbb{N} \rightarrow \mathbb{R}^+$ is a monotonically increasing function with $\lim\limits_{n\rightarrow \infty }φ(n)=+\infty$. We prove that the set $E_{\max}^φ$ is either empty or of full Hausdorff dimension and residual in $[0,1]$ acco…
In this paper we consider diagonally affine, planar IFS $Φ=\left\{S_i(x,y)=(α_ix+t_{i,1},β_iy+t_{i,2})\right\}_{i=1}^m$. Combining the techniques of Hochman and Feng, Hu we compute the Hausdorff dimension of the self-affine attractor and measures and we give an upper bound for the dimension of the exceptional set of parameters.
In this paper, we determine the Hausdorff dimension of the set of points with divergent trajectories on the product of certain homogeneous spaces. The flow is allowed to be weighted with respect to the factors in the product space. The result is derived from its counterpart in Diophantine approximation. In doing this, we introduce a notion of jointly singular matrix tuples, and extend the dimension formula for singular matrices to such matrix tuples.
We extend fundamental results concerning Apollonian packings, which constitute a major object of study in number theory, to certain homogeneous sets that arise naturally in complex dynamics and geometric group theory. In particular, we give an analogue of D. W. Boyd's theorem (relating the curvature distribution function of an Apollonian packing to its exponent and the Hausdorff dimension of the residual set) for Sierpiński carpets that are Julia sets of hyperbolic rational maps.
This paper continues the study of the structure of finite intersections of general multiplicative translates $\mathcal{C}(M_1,\ldots,M_n)=\frac{1}{M_1}Σ_{3,\bar{2}}\cap\cdots\cap\frac{1}{M_n}Σ_{3,\bar{2}}$ for integers $1\leq M_1<\cdots<M_n$, where $Σ_{3,\bar{2}}$ denotes the $3$-adic Cantor set. This study was motivated by questions concerning the discrete dynamical system on the $3$-adic integers $\mathbb{Z}_3$ given by multiplication by $2$. The exceptional set $\mathcal{E}(\mathbb{Z}_3)$ is defined to be the set of all elements of $\mathbb{Z}_3$ whose forward orbits under this action inter…
Singular points for cone actions on the product of certain homogeneous spaces
Institute for Theoretical Sciences, School of Science, Westlake University（西湖大学理学院理论科学研究院）；School of Mathematical Sciences, Peking University（北京大学数学科学学院）
Suppose $g_t$ is a $1$-parameter $\mathrm{Ad}$-diagonalizable subgroup of a Lie group $G$ and $Γ< G$ is a lattice. We study the dimension of bounded and divergent orbits of $g_t$ emanating from a class of curves lying on leaves of the unstable foliation of $g_t$ on the homogeneous space $G/Γ$. We obtain sharp upper bounds on the Hausdorff dimension of divergent on average orbits and show that the set of bounded orbits is winning in the sense of Schmidt (and, hence, has full dimension). The class of curves we study is roughly characterized by being tangent to copies of $\mathrm{SL}(2,\mathbb{R}…
We prove that, after removing a zero Hausdorff dimension exceptional set of parameters, all self-similar measures on the line have a power decay of the Fourier transform at infinity. In the homogeneous case, when all contraction ratios are equal, this is essentially due to Erdős and Kahane. In the non-homogeneous case the difficulty we have to overcome is the apparent lack of convolution structure.
Let X be a h-homogeneous zero-dimensional compact Hausdorff space, i.e. X is a Stone dual of a homogeneous Boolean algebra. It is shown that the universal minimal space M(G) of the topological group G=Homeo(X), is the space of maximal chains on X introduced by Uspenskij. If X is metrizable then clearly X is homeomorphic to the Cantor set and the result was already known (Glasner-Weiss 2003). However many new examples arise for non-metrizable spaces. These include, among others, the generalized Cantor sets X={0,1}^{kappa} for non-countable cardinals kappa, and the corona or remainder of omega, …
Suppose $n\geq 2$ and $\mathcal{A}_{i}\subset \{0,1,\cdots ,(n-1)\}$ for $ i=1,\cdots ,l,$ let $K_{i}=\bigcup\nolimits_{a\in \mathcal{A}_{i}}n^{-1}(K_{i}+a)$ be self-similar sets contained in $[0,1].$ Given $ m_{1},\cdots ,m_{l}\in \mathbb{Z}$ with $\prod\nolimits_{i}m_{i}\neq 0,$ we let \begin{equation*} S_{x}=\left\{ \mathbf{(}y_{1},\cdots ,y_{l}\mathbf{)}:m_{1}y_{1}+\cdots +m_{l}y_{l}=x\text{ with }y_{i}\in K_{i}\text{ }\forall i\right\} . \end{equation*} In this paper, we analyze the Hausdorff dimension and Hausdorff measure of the following set \begin{equation*} U_{r}=\{x:\mathbf{Card}(S_…
