Searcharxiv⌕ Search

arXiv subjects

Xiaobo Hou

Publications and source records attributed to Xiaobo Hou.

13 recordsLinked to original sources
arXiv2026

Conditional entropy realization and approximation by uniquely ergodic measures

This paper studies conditional entropy realization and weak* approximation by uniquely ergodic measures with compact support. We prove that, after fixing an admissible potential average and an entropy strictly below the entropy supremum over the corresponding average fiber, every invariant measure satisfying these two exact constraints can be approximated weakly* by uniquely ergodic measures with compact support satisfying the same constraints. Each approximating measure is the unique invariant measure on its minimal support, whose topological entropy equals the prescribed metric entropy. This result holds for three broad classes of systems: topologically expanding maps (including topologically Anosov systems), transitive countable Markov shifts, and symbolic systems with non-uniform structure. The proof uses a nested multi-horseshoe construction, with separate arguments addressing non-invertibility, non-compactness and non-uniformity.

math.DS↗
arXiv2026

Complete realization of multifractal entropy spectra and pressure functions

We give a complete characterization of the multifractal entropy spectra arising from continuous vector-valued potentials on transitive two-sided shifts of finite type. We prove that, in every finite dimension, any nonnegative upper semicontinuous concave function on a compact convex set that attains the topological entropy as its maximum at a unique point is realized as the entropy spectrum of a potential whose rotation set is precisely that set. Every such spectrum moreover admits arbitrarily many pairwise non-cohomologous realizations. Via Legendre--Fenchel duality, this characterization yields complete pressure flexibility over the entire parameter space. In particular, it resolves the whole-space problem posed by Kucherenko and Quas \cite{KQ2022}. A separate construction based on entropy paths extends scalar spectrum and pressure realization to a substantially broader class of dynamical systems. Finally, with respect to the closed-graph Hausdorff metric, we prove that the spectrum map is lower semicontinuous in every finite dimension, whereas upper semicontinuity fails on a dense set for scalar potentials on transitive shifts of finite type.

math.DS↗
arXiv2026

Analytic Paths of Ergodic Measures with Prescribed Invariants

We study analytic paths of ergodic measures under quantitative constraints. For uniformly hyperbolic systems, we construct one-parameter families of ergodic measures whose prescribed Birkhoff averages vary affinely, whose metric entropies vary analytically, and whose endpoint entropy values are realized exactly. Along these paths, the integral of every Hölder continuous observable depends analytically on the parameter. In the mixing case, the measures may be chosen to be Bernoulli. We also prove a dimension counterpart for average conformal hyperbolic sets: the Hausdorff dimensions of the measures vary analytically along the path. Finally, for a class of partially hyperbolic diffeomorphisms with one-dimensional center, we construct analytic paths of ergodic measures whose center Lyapunov exponents are prescribed linearly and whose entropies vary nalytically.

math.DS↗
arXiv2026

Bohr chaoticity, semi-horseshoes and full-entropy abundance

Bohr chaoticity is a topological notion of dynamical complexity defined through non-orthogonality to all non-trivial weights. It is strictly stronger than positivity of topological entropy and also has strong consequences for the invariant-measure structure. In this paper, we show that every dynamical system having a semi-horseshoe, including every positive-entropy graph map and every $C^1$ partially hyperbolic diffeomorphism, is Bohr chaotic; furthermore, the set of points correlated with any given non-trivial weight has positive topological entropy. Moreover, for positive-entropy dynamical systems with either the shadowing property or the modified almost specification property, such set can has full topological entropy. Our results also yield applications in several classical algebraic and smooth settings, as well as in the $C^0$-generic setting of topological dynamics.

math.DS↗
arXiv2025

Different Statistical Behaviors of Orbits

In this paper, we will study the statistical behaviors of orbits. Firstly, we will show that for a dynamical systems have the shadowing property or almost specification property, the set of nonrecurrent points has full topological entropy. After that, we introduce a criteria for classification of dynamical orbits in order to study the complexity theory of dynamical systems. The criteria is to use upper and lower natural density, upper and lower Banach density to divide different statistical future of dynamical orbits into 56 cases, 28 cases for recurrent orbits and 28 cases for nonrecurrent orbits. We will show the existence of 50 cases and for topologically transitive topologically expanding or topologically transitive topologically Anosov dynamical systems, we will prove that 35 classes, including all the 28 cases for nonrecurrent orbits, can carry full topological entropy. Besides, we will prove that 9 cases can be observable in some differential dynamical systems. Finally, we will apply our results to $\b{eta}-$shifts, $C^{1+α}$ surface diffeomorphisms and Ma$ñ\'$e diffeomorphisms.

math.DS↗
arXiv2024

Topological entropy and Hausdorff dimension of shrinking target sets

In this paper, we study the topological entropy and the Hausdorff dimension of a shrinking target set. We give lower and upper bounds of topological entropy and Hausdorff dimension for dynamical systems with exponential specification property and Lipschitz continuity for maps and homeomorphisms. It generally applies to uniformly hyperbolic systems, expanding systems, and some symbolic dynamics. We show that lower and upper bounds coincide for both topological entropy and Hausdorff dimension when the systems are hyperbolic automorphisms of torus induced from a matrix with only two different eigenvalues, expanding endomorphism of the torus induced from a matrix with only one eigenvalue or some symbolic systems including one or two-sided shifts of finite type and sofic shifts.

math.DS↗
arXiv2024

Abundance of Smale's horseshoes and ergodic measures via multifractal analysis and various quantitative spectrums

In this article, we combine the perspectives of density, entropy, and multifractal analysis to investigate the structure of ergodic measures. We prove that for each transitive topologically Anosov system $(X,f)$, each continuous function $φ$ on $X$ and each $(a,h)\in \mathrm{Int}\{(\int φdμ, h_μ(f)):μ\in M_f(X)\},$ the set $\{μ\in M_f^e(X): (\int φdμ, h_μ(f))=(a,h)\}$ is non-empty and contains a dense $G_δ$ subset of $\{μ\in M_f(X): (\int φdμ, h_μ(f))=(a,h)\}.$ Meanwhile, combining the development of non-hyperbolic systems and cocycles we give a general framework and use it to obtain intermediate entropy property of ergodic measures with same Lyapunov exponent for non-hyperbolic step skew-products, elliptic $\operatorname{SL}(2, \mathbb{R})$ cocycles and robustly non-hyperbolic transitive diffeomorphisms. Moreover, we get generalized results on multiple functions and use them to obtain the intermediate Hausdorff dimension of ergodic measures for transitive average conformal or quasi-conformal Anosov diffeomorphisms, that is $\left\{\operatorname{dim}_H μ: μ\in M_f^e(M)\right\}= \left\{\operatorname{dim}_H μ: μ\in M_f(M)\right\}.$ In this process, we introduce and establish a 'multi-horseshoe' entropy-dense property and use it to get the goal combined with the well-known conditional variational principles. As applications, we also obtain many new observations on various other quantitative spectrums including Lyapunov exponents, first return rate, geometric pressure, unstable Hausdorff dimension, etc.

math.DS↗
arXiv2022

Conditional intermediate entropy and Birkhoff average properties of hyperbolic flows

Katok conjectured that every $C^{2}$ diffeomorphism $f$ on a Riemannian manifold has the intermediate entropy property, that is, for any constant $c \in[0, h_{top}(f))$, there exists an ergodic measure $μ$ of $f$ satisfying $h_μ(f)=c$. In this paper we consider a conditional intermediate metric entropy property and two conditional intermediate Birkhoff average properties for flows. For a basic set $Λ$ of a flow $Φ$ and two continuous function $g,$ $h$ on $Λ,$ we obtain $$\mathrm{Int}\left\{h_μ(Φ):μ\in \mathcal{M}_{erg}(Φ,Λ)\text{ and }\int g dμ=α\right\}=\mathrm{Int}\left\{h_μ(Φ):μ\in \mathcal{M}(Φ,Λ) \text{ and }\int g dμ=α\right\},$$ $$\mathrm{Int}\left\{\int g dμ:μ\in \mathcal{M}_{erg}(Φ,Λ)\text{ and }h_μ(Φ)=c\right\}=\mathrm{Int}\left\{\int g dμ:μ\in \mathcal{M}(Φ,Λ) \text{ and }h_μ(Φ)=c\right\}$$ and $$\mathrm{Int}\left\{\int h dμ:μ\in \mathcal{M}_{erg}(Φ,Λ)\text{ and }\int g dμ=α\right\}=\mathrm{Int}\left\{\int h dμ:μ\in \mathcal{M}(Φ,Λ) \text{ and }\int g dμ=α\right\}$$ for any $α\in \left(\inf_{μ\in \in \mathcal{M}(Φ,Λ) }\int g dμ, \, \sup_{μ\in \in \mathcal{M}(Φ,Λ) }\int g dμ\right)$ and any $c\in (0,h_{top}(Λ)).$ In this process, we establish 'multi-horseshoe' entropy-dense property and use it to get the goal combined with conditional variational principles. We also obtain same result for singular hyperbolic attractors.

math.DS↗
arXiv2021

Entropy of irregular points that are not uniformly hyperbolic

In this article we prove that for a $C^{1+α}$ diffeomorphism on a compact Riemannian manifold, if there is a hyperbolic ergodic measure whose support is not uniformly hyperbolic, then the topological entropy of the set of irregular points that are not uniformly hyperbolic is larger than or equal to the metric entropy of the hyperbolic ergodic measure. In the process of proof, we give an abstract general mechanism to study topological entropy of irregular points provided that the system has a sequence of nondecreasing invariant compact subsets such that every subsystem has shadowing property and is transitive.

math.DS↗
arXiv2021

Ergodic Average Of Typical Orbits And Typical Functions

In this article we mainly aim to know what kind of asymptotic behavior of typical orbits can display. For example, we show in any transitive system, the emprical measures of a typical orbit can cover all emprical measures of dense orbits and can intersect some physical-like measures. In particular, if the union set of emprical measures of all dense orbits is not singleton, then the typical orbit will display historic behavior simultaneously for typical continuous functions and the limit set of ergodic average along every continuous function equals to a closed interval composed by the union of limit sets of ergodic average on all dense orbits. Moreover, if the union set of emprical measures of all dense orbits contains all ergodic measures, the above interval equals to the rotation set. These results are not only suitable for systems with specification-like properties or minimal systems, but also suitable for many other systems including all general (not assumed uniformly hyperbolic) nontrivial homoclinic classes and Bowen eyes. Moreover, we introduce a new property called m-g-product property weaker than classical specification property and minimal property and nontrivial examples are constructed.

math.DS↗
arXiv2021

Strongly distributional chaos of irregular orbits that are not uniformly hyperbolic

In this article we prove that for a diffeomorphism on a compact Riemannian manifold, if there is a nontrival homoclinic class that is not uniformly hyperbolic or the diffeomorphism is a $C^{1+α}$ and there is a hyperbolic ergodic measure whose support is not uniformly hyperbolic, then we find a type of strongly distributional chaos which is stronger than usual distributional chaos and Li-Yorke chaos in the set of irregular orbits that are not uniformly hyperbolic. Meanwhile, we prove that various fractal sets are strongly distributional chaotic, such as irregular sets, level sets, several recurrent level sets of points with different recurrent frequency, and some intersections of these fractal sets. In the process of proof, we give an abstract general mechanism to study strongly distributional chaos provided that the system has a sequence of nondecreasing invariant compact subsets such that every subsystem has exponential specification property, or has exponential shadowing property and transitivity. The advantage of this abstract framework is that it is not only applicable in systems with specification property including transitive Anosov diffeomorphisms, mixing expanding maps, mixing subshifts of finite type and mixing sofic subshifts but also applicable in systems without specification property including $β$-shifts, Katok map, generic systems in the space of robustly transitive diffeomorphisms and generic volume-preserving diffeomorphisms.

math.DS↗
arXiv2021

Topological structures on saturated sets, optimal orbits and equilibrium states

Pfister and Sullivan proved that if a topological dynamical system $(X,T)$ satisfies almost product property and uniform separation property, then for each nonempty compact %convex subset $K$ of invariant measures, the entropy of saturated set $G_{K}$ satisfies \begin{equation}\label{Bowen's topological entropy} h_{top}^{B}(T,G_{K})=\inf\{h(T,μ):μ\in K\}, \end{equation} where $h_{top}^{B}(T,G_{K})$ is Bowen's topological entropy of $T$ on $G_{K}$, and $h(T,μ)$ is the Kolmogorov-Sinai entropy of $μ$. In this paper, we investigate topological complexity of $G_{K}$ by replacing Bowen's topological entropy with upper capacity entropy and packing entropy and obtain the following formulas: \begin{equation*} h_{top}^{UC}(T,G_{K})=h_{top}(T,X)\ \mathrm{and}\ h_{top}^{P}(T,G_{K})=\sup\{h(T,μ):μ\in K\}, \end{equation*} where $h_{top}^{UC}(T,G_{K})$ is the upper capacity entropy of $T$ on $G_{K}$ and $h_{top}^{P}(T,G_{K})$ is the packing entropy of $T$ on $G_{K}.$ In the proof of these two formulas, uniform separation property is unnecessary.

math.DS↗