一个联系自仿射测度与自仿射集的变分原理

Birkhoff平均在自仿射排斥子上的堆积谱

The packing spectrum for Birkhoff averages on a self-affine repeller

本文研究了自仿射Sierpienski海绵上连续势的Birkhoff平均的多重分形分析。特别地，我们给出了一个变分原理的水平集的包装维数。此外，我们证明了填充谱是凹的和连续的。我们给出了填充谱是实解析的一个充分条件，但证明了对于一般的Hölder连续势，情况并非如此.我们还给出了一个精确的标准时，包装谱达到完整的包装尺寸的排斥。再次，我们提出了一个例子，表明这并不总是如此。

Variational principle for weighted topological pressure

Let $π:X\to Y$ be a factor map, where $(X,T)$ and $(Y,S)$ are topological dynamical systems. Let ${\bf a}=(a_1,a_2)\in {\Bbb R}^2$ with $a_1>0$ and $a_2\geq 0$, and $f\in C(X)$. The ${\bf a}$-weighted topological pressure of $f$, denoted by $P^{\bf a}(X, f)$, is defined by resembling the Hausdorff dimension of subsets of self-affine carpets. We prove the following variational principle: $$ P^{\bf a}(X, f)=\sup\left\{a_1h_μ(T)+a_2h_{μ\circπ^{-1}}(S)+\int f \;dμ\right\}, $$ where the supremum is taken over the $T$-invariant measures on $X$. It not only generalizes the variational principle of cl…

Dimensions of random statistically self-affine Sierpinski sponges in $\mathbb R^k$

We compute the Hausdorff dimension of any random statistically self-affine Sierpinski sponge $K\subset \mathbb{R}^k$ ($k\ge 2$) obtained by using some percolation process in $[0,1]^k$. To do so, we first exhibit a Ledrappier-Young type formula for the Hausdorff dimensions of statistically self-affine measures supported on $K$. This formula presents a new feature compared to its deterministic or random dynamical version. Then, we establish a variational principle expressing $\dim_H K$ as the supremum of the Hausdorff dimensions of statistically self-affine measures supported on $K$, and show th…

Unstable Entropies and Variational Principle for Partially Hyperbolic Diffeomorphisms

We study entropies caused by the unstable part of partially hyperbolic systems. We define unstable metric entropy and unstable topological entropy, and establish a variational principle for partially hyperbolic diffeomorphsims, which states that the unstable topological entropy is the supremum of the unstable metric entropy taken over all invariant measures. The unstable metric entropy for an invariant measure is defined as a conditional entropy along unstable manifolds, and it turns out to be the same as that given by Ledrappier-Young, though we do not use increasing partitions. The unstable …

随机动力系统的局部熵理论

Local Entropy Theory of a Random Dynamical System

本文将连续丛随机动力系统的概念推广到$\R$或$\N$的作用被无限可数离散顺从群的作用所取代的情形。给定这样一个系统和一个单调次可加不变随机连续函数族，我们引入了局部纤维拓扑压力的概念，并建立了一个相关的变分原理，将其与测度论熵联系起来.我们还讨论了这个变分原理的一些变形。我们引入连续丛随机动力系统的拓扑和测度熵元组，并应用我们的变分原理来获得这些熵元组之间的关系。最后，我们将这些结果应用到一般的拓扑动力系统中，恢复和推广了局部熵理论中的许多新结果。

Dimensions of "self-affine sponges" invariant under the action of multiplicative integers

Let $m_1 \geq m_2 \geq 2$ be integers. We consider subsets of the product symbolic sequence space $(\{0,\cdots,m_1-1\} \times \{0,\cdots,m_2-1\})^{\mathbb{N}^*}$ that are invariant under the action of the semigroup of multiplicative integers. These sets are defined following Kenyon, Peres and Solomyak and using a fixed integer $q \geq 2$. We compute the Hausdorff and Minkowski dimensions of the projection of these sets onto an affine grid of the unit square. The proof of our Hausdorff dimension formula proceeds via a variational principle over some class of Borel probability measures on the st…

作为开放系统变分原理的Gouy-Stodola定理

Gouy-Stodola Theorem as a variational principle for open systems

近年来对非平衡和远离平衡体系的研究已被证明是有益的，它们在不同学科和许多学科中有着广泛的应用。在科学和工程中，需要一个用独特的分析方法来处理所有这些现象的一般原理：变分原理将具有这种基本作用。本文提出Gouy-Stodola定理作为这一一般的变分原理，既证明了它满足上述要求，又把它与熵产生的统计结果联系起来。

海绵上自仿测度的 $L^q$ 谱

The $L^q$ spectrum of self-affine measures on sponges

本文研究了 $\mathbb{R}^d$ 中海绵上自仿测度的 $L^q$ 谱，通过引入新的压力函数并证明其变分原理，得出了 $L^q$ 谱的变分公式，确定了测度的 Frostman 维数和盒维数，并将现有结果推广至任意维度。

高维自仿系统的Lyapunov指数Hausdorff谱

Hausdorff spectrum of Lyapunov exponents for higher-dimensional self-affine systems

本文研究仿射迭代函数系统Lyapunov指数水平集投影的Hausdorff维数，在典型线性部分条件下给出变分公式，并证明在强开集条件及任意维数下成立。