Dimension estimates for invariant measures of contracting-on-average iterated function systems
Michał Rams
本文对$\R^d$中的平均收缩迭代函数系统的不变测度的维数从上下两个方面进行了估计。
We estimate from above and below the dimension of invariant measure for contracting-on-average iterated function systems in $\R^d$.
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We study contraction conditions for an iterated function system of continuous maps on a metric space which are chosen randomly, identically and independently. We investigate metric changes, preserving the topological structure of the space, which turn the IFS into one which is contracting on average. For the particular case of a system of $C^1$-diffeomorphisms of the circle which is proximal and does not have a probability measure simultaneously invariant by every map, we derive a strongly equivalent metric which contracts on average.
Linear Response for Contracting on Average Iterated Function Systems
研究平均迭代函数系统中线性响应的可微性，分析不同情况下的可微性条件及导数一致性，探讨光滑有界测试函数下不可微的充分条件。
On the Hausdorff dimension of invariant measures of weakly contracting on average measurable IFS
在可分的度量空间中，我们考虑在具有正的、位置依赖概率的可测迭代函数系统下不变的测度。如果测度是遍历的，我们给出了它的豪斯多夫维数的一个上界.我们还证明了它是遍历的当且仅当相关的斜积是。
We analyze the two-point motions of iterated function systems on the unit interval generated by expanding and contracting affine maps, where the expansion and contraction rates are determined by a pair $(M,N)$ of integers. This dynamics depends on the Lyapunov exponent. For a negative Lyapunov exponent we establish synchronization, meaning convergence of orbits with different initial points. For a vanishing Lyapunov exponent we establish intermittency, where orbits are close for a set of iterates of full density, but are intermittently apart. For a positive Lyapunov exponent we show the existe…
Let $\{S_i\}_{i\in Λ}$ be a finite contracting affine iterated function system (IFS) on ${\Bbb R}^d$. Let $(Σ,σ)$ denote the two-sided full shift over the alphabet $Λ$, and $π:Σ\to {\Bbb R}^d$ be the coding map associated with the IFS. We prove that the projection of an ergodic $σ$-invariant measure on $Σ$ under $π$ is always exact dimensional, and its Hausdorff dimension satisfies a Ledrappier-Young type formula. Furthermore, the result extends to average contracting affine IFSs. This completes several previous results and answers a folklore open question in the community of fractals. Some ap…
We study probabilistic iterated function systems (IFS), consisting of a finite or infinite number of average-contracting bi-Lipschitz maps on R^d. If our strong open set condition is also satisfied, we show that both upper and lower bounds for the Hausdorff and packing dimensions of the invariant measure can be found. Both bounds take on the familiar form of ratio of entropy to the Lyapunov exponent.
A topological approach to fuzzy iterated function systems
本文统一 Hutchinson–Barnsley 理论的拓扑与模糊集推广，证明 Tychonoff 空间上压缩 fuzzy IFS 生成模糊吸引子，并推出 Hausdorff 空间中拓扑压缩情形的较弱吸引子结论。
Dimension of contracting on average self-similar measures
本文将Hochman自相似测度维数定理推广至平均收缩测度，证明弱于指数分离的条件即充分，并采用方差求和法避免了熵逆定理的使用。
This paper presents a sufficient condition for a continuum in $R^n$ to be embeddable in $R^n$ in such a way that its image is not an attractor of any iterated function system. An example of a continuum in $R^2$ that is not an attractor of any weakly contracting iterated function system is also given.
Liouville property for groups and conformal dimension
本文证明：若迭代单值群的底层空间共形维数小于2，则该群上所有有限二阶矩对称随机游走具有Liouville性质，从而群可均，并应用于次双曲有理函数的Julia集。
We consider a Markov chain obtained by random iterations of Lipschitz maps $T_i$ chosen with a probability $p_i(x)$ depending on the current position $x$. We assume this system has a property of "contraction on average", that is $\sum_i d(T_ix,T_iy)p_i(x) < ρd(x,y)$ for some $ρ<1$. In the present note, we study the weak convergence of the empirical process associated to this Markov chain.
On absolute continuity of inhomogeneous and contracting on average self-similar measures
本文给出任意维数下自相似测度绝对连续的条件，构造了首例显式绝对连续的非齐次自相似测度，并强化了伯努利卷积及相关结果。
The average shadowing property is considered for set-valued dynamical systems, generated by parameterized IFS, which are uniformly contracting, or conjugacy, or products of such ones. We also prove that if a continuous surjective IFS F on a compact metric space X has the aver- age shadowing property, then every point x is chain recurrent. Moreover, we introduce some examples and investigate the relationship between the original shadowing property and shadowing property for IFS. For example, this is proved that the Sierpinski IFS has the average shadowing property. Then we shows that there is a…
In this note we give a simple sufficient condition for an affine iterated function system to admit an invariant affine subspace persistently with respect to changes in the translation parameters. This yields further examples of tuples of contracting linear maps which do not satisfy the conclusions of Falconer's theorem on the Hausdorff dimension of almost every self-affine set. We also obtain new examples of iterated function systems of similarity transformations which cannot satisfy the open set condition for any choice of translation parameters, and resolve a related question of Peres and So…
The distribution of the largest digit for parabolic Iterated Function Systems of the interval
本文研究了区间抛物线迭代函数系统最大数字的分布，利用非一致扩张 Markov 区间映射的维数理论，证明了相关水平集的 Hausdorff 维数恒等于极限集的维数。
Transversal family of non-autonomous conformal iterated function systems
本文研究具有重叠的非自治共形迭代函数系统，证明满足横截条件的参数族在几乎所有参数下极限集的 Hausdorff 维数等于 m 和 Bowen 维数的最小值，并给出满足横截条件但不满足开集条件的例子。
V-Variable Fractals: Fractals with Partial Self Similarity
我们建立了一种新的分形性质，它在所有尺度上都具有部分自相似性。对于任何具有相关概率分布和任何正整数V的迭代函数系统的集合，存在相应的V-变量分形集或测度类。这些V-变量分形可以从单个确定性迭代函数系统的吸引子上的点得到。在平均压缩性假设下，得到了解的存在唯一性和逼近结果。我们还得到了迭代函数系的一些基本结果的推广。
We show some level-2 large deviation principles for rational maps satisfying a strong form of non-uniform hyperbolicity, called "Topological Collet-Eckmann". More precisely, we prove a large deviation principle for the distribution of iterated preimages, periodic points, and Birkhoff averages. For this purpose we show that each H{ö}lder continuous potential admits a unique equilibrium state, and that the pressure function can be characterized in terms of iterated preimages, periodic points, and Birkhoff averages. Then we use a variant of a general result of Kifer.
We study the dimension of self-similar measures associated to a homogeneous iterated function system of three contracting similarities on $\bf R$ and other more general IFS's. We extend some of the theory recently developed for Bernoulli convolutions to this setting. In the setting of three maps a new phenomenon occurs, which has been highlighted by recent examples of Baker, and Bárány, Käenmäki. To overcome the difficulties stemming form these, we develop novel techniques, including an extension of Hochman's entropy increase method to a function field setup.
We study random independent and identically distributed iterations of functions from an iterated function system of homeomorphisms on the circle which is minimal. We show how such systems can be analyzed in terms of iterated function systems with probabilities which are non-expansive on average.
A Relative Variational Principle for Expanding Iterated Function Systems
本文针对一类扩张迭代函数系，证明了相对变分原理，利用非平稳Ruelle-Perron-Frobenius定理建立了符号基遍历不变测度下边缘熵与诱导斜积纤维平均拓扑熵的关系。
We review the theory of semiattractors associated with non-contractive Iterated Function Systems (IFSs) and demonstrate its applications on a concrete example. In particular, we present criteria for the existence of semiattractors due to Lasota and Myjak. We also discuss the Kieninger criterion which allows us to characterise when a semiattractor is strongly-fibred. Finally, we consider a specific example of a non-contractive IFS introduced by Barnsley and Vince. We find an invariant measure for this system which allows us to describe its semiattractor. The difficulty in analysing this IFS ste…
This paper studies how long it takes the orbit of the chaos game to reach a certain density inside the attractor of a strictly contracting iterated function system of which we only assume that its lower dimension is positive. We show that the rate of growth of this cover time is determined by the Minkowski dimension of the push-forward of the shift invariant measure with exponential decay of correlations driving the chaos game. Moreover, we bound the expected value of the cover time from above and below with multiplicative logarithmic correction terms. As an application, for Bedford-McMullen c…
Let $μ$ be a Borel probability measure associated with an iterated function system consisting of a countably infinite number of contracting similarities and an infinite probability vector. In this paper, we study the quantization dimension of the measure $μ$ with respect to the geometric mean error. The quantization for infinite systems is different from the well-known finite case investigated by Graf and Luschgy. That is, many tools which are used in the finite setting, for example, existence of finite maximal antichains, fail in the infinite case. We prove that the quantization dimension of …
The Minkowski content of a compact set is a fine measure of its geometric scaling. For Lebesgue null sets it measures the decay of the Lebesgue measure of epsilon neighbourhoods of the set. It is well known that self-similar sets, satisfying reasonable separation conditions and non-log comensurable contraction ratios, have a well-defined Minkowski content. When dropping the contraction conditions, the more general notion of average Minkowski content still exists. For random recursive self-similar sets the Minkowski content also exists almost surely, whereas for random homogeneous self-similar …
We study certain multiple ergodic averages of an iterated functions system generated by two contractions on the unit interval. By using the dynamical coding ${0,1}^{\mathbb{N}}$ of the attractor, we compute the Hausdorff dimension of the set of points with a given frequency of the pattern 11 in positions $k, 2k$.
We give a systematic account of iterated function systems (IFS) of weak contractions of different types (Browder, Rakotch, topological). We show that the existence of attractors and asymptotically stable invariant measures, and the validity of the random iteration algorithm ("chaos game"), can be obtained rather easily for weakly contractive systems. We show that the class of attractors of weakly contractive IFSs is essentially wider than the class of classical IFSs' fractals. On the other hand, we show that, in reasonable spaces, a typical compact set is not an attractor of any weakly contrac…
On the fine structure of stationary measures in systems which contract-on-average
设$\{f_1,...,f_m\}$是$\mathbb{R}^d$的一组Lipschitz映射.我们通过独立地选择映射来形成迭代函数系统（IFS），使得以概率$p_i$（$\sum_{i=1}^m p_i=1$）选择映射$f_i$。我们假设IFS平均收缩。我们给出了一个上限的豪斯多夫维数的不变测度上诱导$\mathbb{R}^d$和作为推论表明，该措施将是奇异的，如果模的熵$\sum_i p_i \log p_i$是小于$d$倍的模的系统的李雅普诺夫指数。使用一个版本的香农定理的随机游动半群，我们改进了这一估计，并表明，它实际上是可达到的某些情况下的仿射映射的$\mathbb{R}$。
We establish the equidistribution of the sequence of the averaged pullbacks of a Dirac measure at any value in $\mathbb{C}\setminus\{0\}$ under the derivatives of the iterations of a polynomials $f\in\mathbb{C}[z]$ of degree more than one towards the $f$-equilibrium (or canonical) measure $μ_f$ on $\mathbb{P}^1$. We also show that for every $C^2$ test function on $\mathbb{P}^1$, the convergence is exponentially fast up to a polar subset of exceptional values in $\mathbb{C}$. A parameter space analog of the latter quantitative result for the monic and centered unicritical polynomials family is …
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本文研究了自相似集的自嵌入问题，证明了在共同压缩比和正交部分条件下$f(K)$在$K$中相对开，并部分回答了相关文献中的两个问题。
Empirically defining some constant probabilistic orbits of f(x) and g(x) iterated high-order functions, the stability of these functions in possible entangled interaction dynamics of the environment through its orbit's connectivity (open sets) provides the formation of an exponential dynamic fixed point b = S(n+1) as a metric space (topological property) between both iterated functions for short time lengths. However, the presence of a dynamic fixed point f(g(x)) (x+1) can identify a convergence at iterations for larger time lengths of b (asymptotic stability in Lyapunov sense). Qualitative (Q…
We show the existence of Lebesgue-equivalent conservative and ergodic $σ$-finite invariant measures for a wide class of one-dimensional random maps consisting of piecewise convex maps. We also estimate the size of invariant measures around a small neighborhood of a fixed point where the invariant density functions may diverge. Application covers random intermittent maps with critical points or flat points. We also illustrate that the size of invariant measures tends to infinite for random maps whose right branches exhibit a strongly contracting property on average, so that they have a strong r…
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本文证明满足 UNI 的重叠 C² 迭代函数系统在临界线附近具有一致谱间隙，并由此推出无原子自共形测度的 Fourier 变换多项式衰减及任意重叠下的分形不确定性原理。
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