Jon Chaika
I show that there exist minimal interval exchange transformations with an ergodic measure whose Hausdorff dimension is arbitrarily small, even 0. I will also show that in particular cases one can bound the Hausdorff dimension between $\frac 1 {2r+4}$ and $\frac 1 r$ for any r greater than 1.
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We show some results about the Hausdorff dimension of particular minimal but not uniquely ergodic interval exchange transformations. There is an appendix which shows that typical points for two different ergodic measures of an interval exchange can approximate each other differently.
We show that the set of not uniquely ergodic d-IETs has Hausdorff dimension d-3/2 (in the (d-1)-dimension space of d-IETs) for d>4. For d=4 this was shown by Athreya-Chaika and for d=2,3 the set is known to have dimension d-2.
We show that for almost every (with respect to Masur-Veech measure) $ω\in \mathcal{H}(2)$, the set of angles $θ\in [0, 2π)$ so that $e^{iθ}ω$ has non-uniquely ergodic vertical foliation has Hausdorff dimension (and codimension) $1/2$.
On the Hausdorff Dimension of Measures for a Non-Uniquely Ergodic Family of Interval Exchange Transformations
本文构造了一个具有最大可能测度数的非唯一测度区间交换变换族，并推广了对M.Keane例子中两个测度的Hausdorff维数估计结果。
Hausdorff dimension of non-uniquely ergodic directions in eigenform loci
研究\(\mathcal{H}(1,1)\)中\(D\)非平方时本征形式轨迹\(\mathcal{E}_D\)，证明其中平移曲面\((X,\omega)\)非唯一遍历方向集豪斯多夫维数为\(1/2\)（\(D = 5\)等情况除外）。
In this note, we show that on certain Gatzouras-Lalley carpet, there exist more than one ergodic measures with full Hausdorff dimension. This gives a negative answer to a conjecture of Gatzouras and Peres.
Hausdorff dimension OF dynamical Dophantine approximation associated with ergodic mixing systems
本文针对混合遍历系统，在精确维数与非精确维数条件下分别给出动力覆盖豪斯多夫维数公式，并将圆上二倍映射结果推广至二维环面乘以2、乘以3映射。
In \cite{B} Bourgain proves that Sarnak's disjointness conjecture holds for a certain class of Three-interval exchange maps. In the present paper we slightly improve the Diophantine condition of Bourgain and estimate the constants in the proof. We further show, that the new parameter set has positive, but not full Hausdorff dimension. This, in particular, implies that the Lebesgue measure of this set is zero.
A criterion for unique ergodicity for points of a curve in the space of interval exchange transformation is given.
We show that for odd-valued piecewise-constant skew products over a certain two parameter family of interval exchanges, the skew product is ergodic for a full-measure choice of parameters.
Entropy Density of Ergodic Nonadapted Measures for Markov Interval Maps
本文研究了马尔可夫区间映射中非适应ergodic测度的熵密度性质，证明其在特定度量下为余集且在多数情况下路径连通。
Existence of a Non-Uniquely Ergodic Interval Exchange Transformation with Flips Possessing Three Invariant Measures
本文首次给出了一个具有翻转的区间交换变换（FIET）具有三个不同的不变遍历测度的例子，通过推广M. Keane的方法，利用适应于FIET的Rauzy归纳法，推动了该类动力系统遍历性质的研究。
We consider infinite measure-preserving non-primitive self-similar tiling systems in Euclidean space $\mathbb R^d$. We establish the second-order ergodic theorem for such systems, with exponent equal to the Hausdorff dimension of a graph-directed self-similar set associated with the substitution rule.
Using methods from ergodic theory along with properties of the Furstenberg measure we obtain conditions under which certain classes of plane self-affine sets have Hausdorff or box-counting dimensions equal to their affinity dimension. We exhibit some new specific classes of self-affine sets for which these dimensions are equal.
We consider measures which are invariant under a measurable iterated function system with positive, place-dependent probabilities in a separable metric space. We provide an upper bound of the Hausdorff dimension of such a measure if it is ergodic. We also prove that it is ergodic iff the related skew product is.
Piecewise contracting maps on the interval: Hausdorff dimension, entropy and attractors
本文研究紧区间上单射分段压缩映射的吸引子，利用轨道复杂度证明其拓扑熵与豪斯多夫维数均为零。
We give conditions for minimality of $\mathbb Z/N\mathbb Z$ extensions of a rotation of angle $α$ with one marked point, solving the problem for any prime $N$: for $N=2$, these correspond to the Veech 1969 examples, for which a necessary and sufficient condition was not known yet. We provide also a word combinatorial criterion of minimality valid for general interval exchange transformations, which applies to $\mathbb Z/N\mathbb Z$ extensions of any interval exchange transformation with any number of marked points. Then we give a condition for unique ergodicity of these extensions when the ini…
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…
We prove linear upper and lower bounds for the Hausdorff dimension set of minimal interval exchange transformations with flips (in particular without periodic points), and a linear lower bound for the Hausdorff dimension of the set of non-uniquely ergodic minimal interval exchange transformations with flips.
This paper first studies the measure theoretic pressure of measures that are not necessarily ergodic. We define the measure theoretic pressure of an invariant measure (not necessarily ergodic) via the Carathéodory-Pesin structure described in \cite{Pes97}, and show that this quantity is equal to the essential supremum of the free energy of the measures in an ergodic decomposition. To the best of our knowledge, this formula is new even for entropy. Meanwhile, we define the measure theoretic pressure in another way by using separated sets, it is showed that this quantity is exactly the free ener…
We show that the Hausdorff dimension of any proper Teichmüller horocycle flow orbit closure on any $\mathrm{SL}{(2,\mathbf{R})}$-invariant subvariety of Abelian or quadratic differentials is bounded away from the dimension of the subvariety in terms of the polynomial mixing rate of the Teichmüller horocycle flow on the subvariety. The proof is based on abstract methods for measurable flows adapted from work of Bourgain and Katz on sparse ergodic theorems.
We study the existence of transitive exchange maps with flips defined on the unit circle. We provide a complete answer to the question of whether there exists a transitive exchange map of the unit circle defined on n subintervals and having f flips.
Ergodic properties of infinite extension of symmetric interval exchange transformations
本文研究了对称区间交换变换的无限扩张的遍历性质，证明了特定余循环的斜积在遍历基下是遍历的，并在唯一遍历性假设下推广到更一般的函数，最后通过弱混合情形得到无限遍历指数。
Expanding Maps on Flowers, Interval Exchange Transformations, and Ergodic Optimization
研究花状集的扩张映射性质，探讨其与区间交换变换的关系，并通过数值结果支持三角多项式在花状集上最大化的猜想。
On the Hausdorff dimension of geodesics that diverge on average
本文证明了在完备捏缩负曲率黎曼流形上，回复且平均发散的测地线方向的豪斯多夫维数等于测地流在无穷远处的熵，并给出无限遍历保守不变测度熵的上界。
Invariant measures with full support and approximation by zero-entropy systems in the $C^0$-Gromov--Hausdorff topology
本文证明紧度量空间上连续映射在$C^0$-Gromov--Hausdorff拓扑下可由零熵映射逼近，依赖遍历分解定理和正轨道点稠密性，揭示拓扑熵不稳定性及周期点稠密性，最终得出全支撑不变测度动态落入零熵系统逼近范围。
We compute the dimension spectrum of certain nonconventional averages, namely, the Hausdorff dimension of the set of $0,1$ sequences, for which the frequency of the pattern 11 in positions $k, 2k$ equals a given number $θ\in [0,1]$.
We prove that the skew product over a linearly recurrent interval exchange transformation defined by almost any real-valued, mean-zero linear combination of characteristic functions of intervals is ergodic with respect to Lebesgue measure.
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.
We consider the escaping parameters in the family $β\wp_Λ$, i.e. these parameters for which the orbits of critical values of $β\wp_Λ$ approach infinity, where $\wp_Λ$ is the Weierstrass function. Unlike to the exponential map the considered functions are ergodic. They admit a non-atomic, $σ$-finite, ergodic, conservative and invariant measure $μ$ absolutely continuous with respect to the Lebesgue measure. Under additional assumptions on the $\wp_Λ$-function we estimate from below the Hausdorff dimension of the set of escaping parameters in the family $β\wp_Λ$, and compare it with the Hausdorff…
Hausdorff Dimension and Flexibility for Attractors of Non-Uniformly Hyperbolic Flows
该研究将几何洛伦兹吸引子的维数理论推广到更一般的非均匀双曲吸引子，证明此类吸引子含维数大于2的双曲马蹄，建立中间维数性质，得到相关动力学谱的鲁棒非空内部结果。
We show that for any ergodic Lebesgue measure preserving transformation $f: [0,1) \rightarrow [0,1)$ and any decreasing sequence $\{b_i\}_{i=1}^{\infty}$ of positive real numbers with divergent sum, the set $$\underset{n=1}{\overset{\infty}{\cap}} \, \underset{i=n}{\overset{\infty}{\cup}}\, f^{-i}(B (R_α^{i} x,b_i))$$ has full Lebesgue measure for almost every $x \in [0,1)$ and almost every $α\in [0,1)$. Here $B(x,r)$ is the ball of radius $r$ centered at $x \in [0,1)$ and $R_α: [0,1) \rightarrow [0,1)$ is rotation by $α\in [0,1)$. As a corollary, we provide partial answer to a question asked …
For integer $m\ge3$, we study the dynamical system $(Λ_m,σ_m)$ where $Λ_m$ is the set $\{w\in\{0,1\}^\mathbb{N}: w$ does not contain $0^m$ or $1^m\}$ and $σ_m$ is the shift map on $\{0,1\}^\mathbb{N}$ restricted to $Λ_m$, study the Bernoulli-type measures on $Λ_m$ and find out the unique equivalent $σ_m$-invariant ergodic probability measure. As an application, we obtain the Hausdorff dimension of the set of univoque sequences, the Hausdorff dimension of the set of sequences in which the lengths of consecutive $0$'s and consecutive $1$'s are bounded, and the Hausdorff dimension of their freque…
We show that there exists an interval exchange and a point so that the orbit of the point equidistributes for a measure that is not ergodic.
We construct a self-affine sponge in $\mathbb R^3$ whose dynamical dimension, i.e. the supremum of the Hausdorff dimensions of its invariant measures, is strictly less than its Hausdorff dimension. This resolves a long-standing open problem in the dimension theory of dynamical systems, namely whether every expanding repeller has an ergodic invariant measure of full Hausdorff dimension. More generally we compute the Hausdorff and dynamical dimensions of a large class of self-affine sponges, a problem that previous techniques could only solve in two dimensions. The Hausdorff and dynamical dimens…
In this work we present an example of C^\infty-diffeomorphism of a compact 4-manifold such that it admits a global SRB measure μbut for which the special ergodic theorem doesn't hold. Namely, for this transformation there exist a continuous function ϕand a positive constant αsuch that the following holds: the set of the initial points for which the Birkhoff time averages of the function ϕdiffer from its μ--space average by at least αhas zero Lebesgue measure but full Hausdorff dimension.
We show the existence of a bounded Borel measurable saturated compensation function for a factor map between subshifts. As an application, we find the Hausdorff dimension and measures of full Hausdorff dimension for a compact invariant set of an expanding nonconformal map on the torus given by an integer-valued diagonal matrix. These problems were studied in [19] for a compact invariant set whose symbolic representation is a shift of finite type under the condition of the existence of a saturated compensation function. We extend the results by presenting a formula for the Hausdorff dimension f…
We study the ergodic properties of compositions of interval exchange transformations and rotations. We show that for any interval exchange transformation T, there is a full measure set of αin [0, 1) so that T composed with R_α is uniquely ergodic, where R_α is rotation by α.
The Dimension of Integral Self-Affine Sets via Fractal Perturbations: The Box and the Hausdorff Dimensions, Ergodic Measures
本文通过分形扰动方法研究积分自仿集的维度问题，推导出其维数公式，并证明了其盒维数的存在性，同时展示了全维的测度存在性。
Exchange of Intervals and Intrinsic Ergodicity of the Negative Beta shift
研究负贝塔移位的内在ergodic性，探讨代数整数序列与最大熵测度的关系，揭示黄金比例与贝塔值的关联。
We discuss limit distributions for hitting-time functions of certain exceptional families of asymptotically rare events for ergodic probability preserving transformations. The abstract core is an inducing argument. The latter applies, for example, to shrinking intervals around periodic points (both uniformly expanding and neutral) of certain finite measure preserving interval maps. In particular, we give a complete answer to a question raised in [FFTV].
We prove a spatial limit theorem for generic interval exchange transformations (IETs): for a generic IET the normalized ergodic sums of a sufficiently regular (e.g., Lipschitz) function have the same asymptotic behavior of distributions as the behaviour of ergodic integrals for a generic translation flow on a flat surface, described by A. Bufetov.
On the Hausdorff dimension of invariant measures of piecewise smooth circle homeomorphisms
本文证明具有无理旋转数和零平均非线性的充分正则分段光滑圆同胚的唯一不变测度通常具有零Hausdorff维数，并借助广义区间交换变换的组合旋转数概念编码该一般性条件。
On measures and semiconjugacies for affine interval exchange transformations
本文研究双曲周期型仿射区间交换变换的不变测度与共形测度的豪斯多夫维数，给出中心稳定型条件下的精确公式，并推导周期情形下半共轭的最高 Hölder 指数显式公式。
We consider interval exchange transformations of periodic type and construct different classes of recurrent ergodic cocycles of dimension $\geq 1$ over this special class of IETs. Then using Poincaré sections we apply this construction to obtain recurrence and ergodicity for some smooth flows on non-compact manifolds which are extensions of multivalued Hamiltonian flows on compact surfaces.
We give conditions that characterize the existence of an absolutely continuous invariant probability measure for a degree one $C^2$ endomorphism of the circle which is bimodal, such that all its periodic orbits are repelling, and such that both boundaries of its rotation interval are irrational numbers. Those conditions are satisfied when the boundary points of the rotation interval belong to a Diophantine class. In particular they hold for Lebesgue almost every rotation interval. The measure obtained is a global physical measure, and it is hyperbolic.
We discuss on some families of skew product maps on a square. For a kind of skew product maps with coupled-expanding property, we estimate Hausdorff dimension of its attractor. And we prove that there exists an ergodic measure with full Hausdorff dimension for this kind of skew product maps.
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.
We show that for every linear toral automorphism, especially the non-hyperbolic ones, the entropies of ergodic measures form a dense set on the interval from zero to the topological entropy.
We prove that irreducible, linearly recurrent, type W interval exchange transformations are always mild mixing. For every irreducible permutation the set of linearly recurrent interval exchange transformations has full Hausdorff dimension.
Thanks to works by M. Kontsevich and A. Zorich followed by C. Boissy, we have a classification of all Rauzy Classes of any given genus. It follows from these works that Rauzy Classes are closed under the operation of inverting the permutation. In this paper, we shall prove the existence of self-inverse permutations in every Rauzy Class by giving an explicit construction of such an element satisfying the sufficient conditions. We will also show that self-inverse permutations are Lagrangian, meaning any suspension has its vertical cycles span a Lagrangian subspace in homology. This will simplify…
We study differentiability of topological conjugacies between expanding piecewise $C^{1+ε}$ interval maps. If these conjugacies are not $C^1$, then they have zero derivative almost everywhere. We obtain the result that in this case the Hausdorff dimension of the set of points for which the derivative of the conjugacy does not exist lies strictly between zero and one. Using multifractal analysis and thermodynamic formalism, we show that this Hausdorff dimension is explicitly determined by the Lyapunov spectrum. Moreover, we show that these results give rise to a "rigidity dichotomy" for the typ…
We investigate the set of biaccessible points for connected polynomial Julia sets of arbitrary degrees $d\geq 2$. We prove that the Hausdorff dimension of the set of external angles corresponding to biaccessible points is less than 1, unless the Julia set is an interval. This strengthens theorems of Stanislav Smirnov and Anna Zdunik: they proved that the same set of external angles has zero 1-dimensional measure.
We consider piecewise monotone maps, we show that an ergodic measure for which the map is invertible almost everywhere can not be mixing. It follows that every ergodic measure for an interval translation mapping is not mixing. We also show that double rotations without periodic points have an ergodic but not weakly mixing invariant measure. This article is dedicated to the memory of Anatoly Mikhailovich Stepin.
Ergodic measures for periodic type $\mathbb{Z}^m$-skew-products over Interval Exchange Transformations
本文研究了区间交换变换上周期型$\mathbb{Z}^m$-斜积的遍历不变Radon测度分类问题，通过引入adic编码的符号语言并应用相关符号结果，证明了此类测度恰为由$\mathbb{R}^m$参数化的Maharam测度，并给出了其显式形式与弱*-连续性。
We propose an extension of ergodic theory which focuses on the identification of ergodicity in terms of the uniqueness of the invariant measure. We first explain the concept for the doubling maps, which can be analyzed using Fourier methods. We then proceed to the Gauss-type maps of interest, of the form $x\mapsto -β/x$ mod $2\mathbb Z$ on the symmetric interval $[-1,1]$, for $0<β\le1$. We study an extended state space on the interval, formed as the restriction to the interval $[-1,1]$ of functions of the form $f+\mathbf{H}g$, where $f$ and $g$ are $L^1$-functions. We then look for invariant s…
In this paper, we study the Hausdorff dimension of the generalized intrinsic level set with respect to the given ergodic meausre in a class of non-uniformly hyperbolic interval maps with finitely many branches.
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…
This work is devoted to the study of families of infinite parabolic iterated function systems (PIFS) on a closed interval parametrized by vectors in $\mathbb{R}^d$ with overlaps. We show that the Hausdorff dimension and absolute continuity of ergodic projections through a family of infinite PIFS are decided \emph{a.e.} by the growth rate of the entropy and Lyapunov exponents of the families of truncated PIFS with respect to the concentrating measures, under transversality of the family essentially. We also give an estimation on the upper bound of the Hausdorff dimension of parameters where the…
In this paper, we prove a criterion for existence of continuous non constant eigenfunctions for interval exchange transformations, that is for non topologically weak mixing. We first construct, for any m>3, uniquely ergodic interval exchange transformations of rank 2 with irrational eigenvalues associated to continuous eigenfunctions, so that are not topologically weak mixing, this answers to a question of Ferenczi and Zamboni. Then, we construct, for any even number m>3, interval exchange transformations of rank 2 with both irrational eigenvalues (associated to continuous eigenfunctions) and …
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