The Manneville map: topological, metric and algorithmic entropy

Fiber entropy and algorithmic complexity of random orbits

Let $Θ$ be a finite alphabet. We consider a bundle of measure preserving transformations $(T_θ)_{θ\in Θ}$ acting on a probability space $(X,μ)$, which are chosen randomly according to an ergodic stochastic process $(Ξ,ν,σ)$ with state space $Θ$. This describes a paradigmatic case of a random dynamical system (RDS). Considering a finite partition $\mathcal{P}$ of $X$ we show that the conditional algorithmic complexity of a random orbit $x, T_{α_{0}}(x),T_{α_{1}}\circ T_{α_{0}}(x),...$ in $X$ along a sequence $α= α_{0}α_{1}α_{2}...$ in $Ξ$ equals almost surely the fiber entropy of the RDS with r…

通过紧型子移位研究度量平均维数

Metric mean dimension via subshifts of compact type

该研究针对紧型子移位的度量平均维数展开，证明连续映射与其逆极限的度量平均维数一致以推广Bowen熵公式，还将该概念扩展至不连续映射，并验证了Gauss映射等的度量平均维数与盒维数、压力算子临界参数的等价性。

The Algorithmic Information Content for randomly perturbed systems

In this paper we prove estimates on the behaviour of the Kolmogorov-Sinai entropy relative to a partition for randomly perturbed dynamical systems. Our estimates use the entropy for the unperturbed system and are obtained using the notion of Algorithmic Information Content. The main result is an extension of known results to study time series obtained by the observation of real systems.

一致测试与算法热力学熵

Uniform Tests and Algorithmic Thermodynamic Entropy

本文证明了可计算度量空间中通用一致测试分数的下界关系，并据此推导算法热力学熵在动力学存在时必然振荡，且离群点会在可计算动力学中涌现。

Algorithmic information for intermittent systems with an indifferent fixed point

Measuring the average information that is necessary to describe the behaviour of a dynamical system leads to a generalization of the Kolmogorov-Sinai entropy. This is particularly interesting when the system has null entropy and the information increases less than linearly with respect to time. We consider two classes of maps of the interval with an indifferent fixed point at the origin and an infinite natural invariant measure. We calculate that the average information that is necessary to describe the behaviour of its orbits increases with time $n$ approximately as $n^α$, where $α<1$ depends…

Entropy stability and Milnor-Thurston invariants for Bowen-Series-like maps

We define a family of discontinuous maps on the circle, called Bowen-Series-like maps, for geometric presentations of surface groups. The family has $2N$ parameters, where $2N$ is the number of generators of the presentation. We prove that all maps in the family have the same topological entropy, which coincides with the volume entropy of the group presentation. This approach allows a simple algorithmic computation of the volume entropy from the presentation only, using the Milnor-Thurston theory for one dimensional maps.