arXiv subjects

Xiaoyao Zhou

Publications and source records attributed to Xiaoyao Zhou.

At least 19 recordsLinked to original sources

arXiv2026

Variational Principles for $r$-Neutralized Pfister-Sullivan Entropy

In this paper, we study $r$-neutralized Pfister--Sullivan entropy, defined by measuring the complexity of orbit segments whose empirical measures lie near a prescribed invariant measure. We mainly obtain the following three results: (i) We disprove a variational conjecture posed by C.~Dong and Q.~Qiao [On $r$-neutralized entropy: entropy formula and existence of measures attaining the supremum, Comm. Math. Phys. 406 (2025), Article No. 74.] (ii)] We obtain variational formulas over probability measures for both upper and lower $r$-neutralized topological entropies. We also introduce new measure-theoretic quantities through PS localization and establish some variational principles for upper $r$-neutralized topological entropy (iii) We obtain upper bounds for $r$-neutralized PS entropy under different smoothness assumptions and lower bounds for $C^{1+α}$ diffeomorphisms, in terms of entropy, dimension and Lyapunov exponents. For $C^\infty$ diffeomorphisms and $C^{1}$ Anosov diffeomorphisms, the $r$-neutralized PS entropies equal $h_μ(f)+r\dim M$ for every ergodic measure $μ$. We also establish exact formulas on mixing surface hyperbolic basic sets and show that the whole-manifold measure formula can fail for $C^2$ surface diffeomorphisms.

math.DS↗

arXiv2026

Maximizing Families and Typical Periodic Optimization for Almost Additive Sequences

In this paper, we study typical periodic optimization (TPO) for almost additive potentials in two perturbation spaces. Our main setting is the Banach quotient $\mathcal E_{\rm orb}(X,T)$ of orbit-Lipschitz almost additive potentials, where we extend the theory of maximizable sets and countable maximizable families developed by W. Huang, O. Jenkinson, L. Xu and Y. Zhang [Typical periodic optimization for dynamical systems: symbolic dynamics, Invent. Math. 245 (2026), 1--63], and establish a global structural theorem. For a countable maximizable family, global TPO holds if every non boundary member has $X$-extendable TPO and the boundary region has empty interior. As an application, we construct a compact system with global TPO for which $\mathcal E_{\rm orb}(X,T)$ is infinite-dimensional and the maximizing periods in open locking regions are unbounded. For a fixed almost additive potential $Φ$, we also develop relative TPO theory on its Lipschitz leaf. When $Φ=0$, this framework reduces to classical Lipschitz TPO. We prove the corresponding leafwise structural theorem and give a non additive rank-one matrix example on a full shift.

math.DS↗

arXiv2026

Entropy Scales in Topological Dynamical Systems

Motivated by Helfter's notion of scaling, we define Bowen, upper capacity, and local measure-theoretic entropy scales. Under a controlled decay condition, we establish a variational principle on compact subsets by combining a Billingsley-type theorem with a Frostman-type construction. We also prove factor inequalities for upper capacity entropy scales on compact sets and for Bowen entropy scales on arbitrary subsets. For induced systems on spaces of probability measures, we give sufficient conditions for the preservation of zero upper capacity entropy scales and show, under an additional comparison condition, that positivity for the original system forces the induced entropy scale to be infinite. Finally, we define upper capacity entropy scales along prescribed observation times and establish the corresponding zero-level equivalence for induced systems.

math.DS↗

arXiv2026

Katok's intermediate entropy conjecture for amenable group actions

In this paper, we investigate a broad class of systems for which Katok's intermediate entropy conjecture holds. In particular, we show that a dynamical system with an amenable group action satisfies Katok's intermediate entropy conjecture provided that it has the specification property and is asymptotically entropy expansive.

math.DS↗

arXiv2026

Global Survey of Technologies and Industrial Applications of Grid Forming Energy Storage Systems

Grid-forming (GFM) energy storage system (ESS) is a key enabler for stabilizing future power systems with high penetration of converter-based resources (CBRs). To get a better overview of the state-of-the-art and challenges for implementing and deploying GFM-ESS, a global survey has been initiated by Cigre Working Group B4.101 - industrial implementation and application of grid forming energy storage systems. Feedback was collected from universities, transmission system operators (TSOs), power plant developers, original equipment manufacturers (OEMs), research institutes, as well as consultants. It is interesting to note that while many common understandings have been established in practice, certain gaps persist among different stakeholders. This article intends to bridge this gap by presenting a summary of the survey, including the questionnaire, responses from various stakeholders, and in-depth analysis of the survey results. The key challenges faced by different stakeholders in deploying GFM-ESS are identified, shedding light on future research in this direction.

eess.SY↗

arXiv2026

Ergodic measures of intermediate entropies for $\mathbb{Z}^{d}$-action

For dynamical systems satisfying the approximate $\mathbb{Z}^{d}$ or $\mathbb{Z}_+^{d}$-product property and asymptotically entropy expansiveness, we establish a precise description of the structure of their space of invariant measures. In particular, we prove that the set of ergodic measures with any given intermediate entropy is generic in certain natural subspaces. As a consequence, this result confirms Katok's conjecture on the existence of ergodic measures with arbitrary intermediate entropy for such systems.

math.DS↗

arXiv2026

Variational principle for random pressure function

For random dynamical systems, by summarizing the fundamental properties of Kifer's topological pressure we introduce the concept of random pressure functions, and define Ruelle's metric entropy for invariant measures. Employing the techniques from convex analysis and ergodic theory, we establish a variational principle for random pressure functions. Consequently, this new variational principle allows us to establish a vital bridge between ergodic theory and topological dynamics. In particular, the variational principles for polynomial topological entropy in zero entropy systems, mean dimensions in infinite entropy systems, and preimage entropy-like quantities in non-invertible dynamical systems are obtained.

math.DS↗

arXiv2025

Metric mean dimension and the variational principle for actions of amenable groups

Metric mean dimension is a dynamical counterpart of the box dimension in fractal geometry to characterize the topological complexity of infinite entropy systems. The classical variational principle states that topological entropy equals the supremum of measure-theoretic entropy over the set of invariant measures. Lindenstrauss and Tsukamoto proved that this variational principle fails for metric mean dimension in terms of rate-distortion dimensions. For the actions of amenable groups, we define a new measure-theoretic metric mean dimension for invariant measures and establish a classical-type variational principle for metric mean dimension. In particular, we extend the Lindenstrauss-Tsukamoto variational principles to the classical variational principle by defining modified rate-distortion dimensions. As applications, for systems with zero metric mean dimension, we introduce infinite entropy dimensions in both topological and measure-theoretic settings, and relate them via a variational principle. For systems with positive metric mean dimension, we introduce local metric mean dimension from a local perspective, and relate it to the metric mean dimension of the whole phase space via variational principles.

math.DS↗

arXiv2025

$α$-BS dimension on subsets

We aim to investigate the dimension theory of $α$-pressure-like quantities. By means of the Carath$\acute{\rm e}$odory-Pesin structure, we define $α$-BS dimension and $α$-Pesin topological pressure on subsets using $α$-Bowen metric $$d_{n}^α(x,y)=\max_{0\leq i\leq n-1}e^{αi}d(f^{i}x,f^{i}y),$$ where $α\geq 0$. Specifically, we show that $α$-BS dimension and $α$-Pesin topological pressure are related by a Bowen's equation. Inspired by the classical Brin-Katok entropy, we introduce the notion of $α$-local Brin-Katok entropy, and establish a variational principle for $α$-BS dimension on compact subsets in terms of $α$-local Brin-Katok entropy. Besides, for subshifts of finite type, we prove that $α$-Bowen topological entropy is closely related to spectral radius and Hausdorff dimension.

math.DS↗

arXiv2025

Bowen's equations for invariance pressure of control systems

We aim to establish Bowen's equations for upper capacity invariance pressure and Pesin-Pitskel invariance pressure of discrete-time control systems. We first introduce a new invariance pressure called induced invariance pressure on partitions that specializes the upper capacity invariance pressure on partitions, and then show that the two types of invariance pressures are related by a Bowen's equation. Besides, to establish Bowen's equation for Pesin-Pitskel invariance pressure on partitions we also introduce a new notion called BS invariance dimension on subsets. Moreover, a variational principle for BS invariance dimension on subsets is established.

math.OC↗

arXiv2024

On variational principle for upper metric mean dimension with potential

Borrowing the idea of topological pressure determining measure-theoretical entropy in topological dynamical systems, we establish a variational principle for upper metric mean dimension with potential in terms of upper measure-theoretical metric mean dimension of invariant measures. Moreover, the notion of equilibrium state is introduced to characterize these measures that attain the supremum of the variational principle.

math.DS↗

arXiv2024

Non-dense orbits on topological dynamical systems

Let $(X,d,T )$ be a topological dynamical system with the specification property. We consider the non-dense orbit set $E(z_0)$ and show that for any non-transitive point $z_0\in X$, this set $E(z_0)$ is empty or carries full topological pressure.

math.DS↗

arXiv2024

Upper metric mean dimensions with potential of $ε$-stable sets

It is well-known that $ε$-stable sets have a deep connection with the topological entropy of dynamical systems. In the present paper, we investigate the relationships of three types of upper metric mean dimensions with potential between \emph{the blocks of $ε$-stable sets, $ε$-stable sets, the dispersion of preimages of $ε$-stable sets} and the whole phase space. Besides, some chaotic phenomenons are revealed in infinite entropy systems. As an application of main results, we show tail entropy, preimage neighborhood entropy and topological entropy have the same metric mean dimension.

math.DS↗

arXiv2024

Measure-theoretic metric mean dimension

For infinite measure-theoretic entropy systems, we introduce the notion of measure-theoretic metric mean dimension of invariant measures for different types of measure-theoretic $ε$-entropies, and show that measure-theoretic metric mean dimensions of different types of measure-theoretic $ε$-entropies coincide with the packing metric mean dimension of the set of generic points of ergodic measures.

math.DS↗

arXiv2024

Irregular set and metric mean dimension with potential

Let $(X,f)$ be a dynamical system with the specification property and $φ$ be a continuous function. In this paper, we consider the multifractal irregular set \begin{align*} I_φ=\left\{x\in X:\lim\limits_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}φ(f^ix)\ \text{does not exist}\right\} \end{align*} and show that this set is either empty or carries full Bowen upper and lower metric mean dimension with potential.

math.DS↗

arXiv2024

Multifractal level sets and metric mean dimension with potential

Let $(X,f)$ be a dynamical system with the specification property and $φ$ be continuous functions. In this paper, we establish some conditional variational principles for the upper and lower Bowen/packing metric mean dimension with potential of multifractal level set $K_α:=\{x\in X:\lim\limits_{n\to\infty}\dfrac{1}{n}\sum\limits_{i=0}^{n-1}φ(f^ix)=α\}.$

math.DS↗

arXiv2024

Packing topological pressure for amenable group actions

In this paper, we first prove the variational principle for amenable packing topological pressure. Then we obtain an inequality concerning amenable packing pressure for factor maps. Finally, we show that the equality about packing topological pressure of the set of generic points when the system satisfies the almost specification property, or $μ$ is ergodic.

math.DS↗

arXiv2024

Variational principle for neutralized Bowen topological entropy

Ovadia and Rodriguez-Hertz defined neutralized Bowen open ball as $$B_n(x,e^{-nε})=\{y\in X: d(T^jx, T^jy)<e^{-nε}, \forall 0\leq j\leq n-1\}.$$ We introduce the notion of neutralized Bowen topological entropy of subsets by neutralized Bowen open ball, and establish variational principles for neutralized Bowen topological entropy of compact subsets in terms of neutralized Brin-Katok local entropy and neutralized Katok's entropy.

math.DS↗