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Xueting Tian

Publications and source records attributed to Xueting Tian.

At least 19 recordsLinked to original sources
arXiv2026

Variations of topological theory and ergodic theory via gap function in non-uniform specification

In our previous work [43], we studied qualitative differences between specification and nonuniform specification. In this paper, we investigate quantitative variations depending on the gap function. We first obtain lower bounds for the Bowen topological entropy of irregular sets in terms of the lower linear growth of gap function. In contrast, we prove that every non-empty over-saturated set has full packing topological entropy under non-uniform specification. We also establish a quantitative lower bound for the Bowen topological entropy of transitive points under non-uniform specification, and a lower bound for the exponential growth of periodic orbits under its periodic version. Besides, we construct symbolic systems with non-uniform specification with a given gap growth rate which contain an arbitrary subshift. These systems show that certain bounds concerning irregular sets and periodic orbits are optimal. They also show that positive linear gap growth may destroy full Bowen entropy of transitive points, the conditional variational principle, the intermediate entropy and pressure properties, and the genericity of continuous functions whose unique maximizing measure has zero entropy.

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arXiv2026

Realizing prescribed entropy functions by smooth diffeomorphisms of closed manifolds of dimension at least three

Let $M$ be a closed smooth manifold of dimension $d\geq3$. Given a compact metrizable Choquet simplex $\mathscr S$ and a bounded nonnegative affine upper semicontinuous function $\mathfrak e$ on $\mathscr S$, we construct a $C^\infty$ diffeomorphism $h$ of $M$, isotopic to $\operatorname{id}_M$ and supported in an embedded $d$-dimensional solid torus $D^{d-1}\times S^1$, with an isolated minimal invariant Cantor set $K$. The invariant-measure simplex of $h|_K$ is affinely homeomorphic to $\mathscr S$ with entropy function $\mathfrak e$, whereas every ergodic $h$-invariant measure not supported on $K$ is a Dirac measure at a fixed point. Consequently, the set of measure-theoretic entropies of ergodic $h$-invariant probability measures and the topological entropy of $h$ are \[ \mathscr H_{\mathrm e}(h)=\{0\}\cup\mathfrak e(\operatorname{ex}\mathscr S), \qquad h_{\mathrm{top}}(h)=\max_{p\in\mathscr S}\mathfrak e(p). \] The map $h$ is $C^\infty$-approximable by zero-entropy diffeomorphisms isotopic to $\operatorname{id}_M$. Taking $\mathscr S$ to be a singleton yields counterexamples to Katok's intermediate-entropy conjecture on every such $M$. We also construct such counterexamples $h_j$ and numbers $c_j>0$ with $h_j\to\operatorname{id}_M$ in $C^\infty$, $c_j\to0$, and \[ \mathscr H_{\mathrm e}(h_j)=\{0,c_j\}, \qquad h_{\mathrm{top}}(h_j)=c_j. \] Hence the intermediate-entropy property is not $C^\infty$ open among diffeomorphisms isotopic to the identity.

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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.

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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.

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arXiv2026

On the loss of upper semi-continuity of metric entropy for $C^{r}$ diffeomorphisms

In this article, we give an upper bound estimate for the quantitative loss of upper semicontinuity of metric entropy for $C^r\:(r>1)$ diffeomorphisms. Building on earlier entropy estimates and reparametrization methods, we optimize the upper bound estimate with respect to both dimension and asymptotic Lipschitz constant. Motivated by examples of Newhouse and Buzzi, we show that the estimate is sharp.

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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.

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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.

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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.

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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.

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arXiv2024

A $C^r$-connecting lemma for Lorenz attractors and its application on the space of ergodic measures

For every $r\in\mathbb{N}_{\geq 2}\cup\{\infty\}$, we prove a $C^r$-connecting lemma for Lorenz attractors. To be precise, for a Lorenz attractor of a $3$-dimensional $C^r$ ($r\geq 2$) vector field, a heteroclinic orbit associated to the singularity and a critical element can be created through arbitrarily small $C^r$-perturbations. As an application, we show that for $C^r$-dense geometric Lorenz attractors, the Dirac measure of the singularity is isolated inside the space of ergodic measures and thus the ergodic measure space is not connected; while for $C^r$-generic geometric Lorenz attractors, the space of ergodic measures is path connected with dense periodic measures. In particular, the generic part proves a conjecture proposed by C. Bonatti in $C^r$-topology for Lorenz attractors.

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arXiv2024

Non-uniform Cocycles for Some Uniquely Ergodic Minimal Dynamical Systems on Connected Spaces

In this paper, we pay attention to a weaker version of Walters's question on the existence of non-uniform cocycles for uniquely ergodic minimal dynamical systems on non-degenerate connected spaces. We will classify such dynamical systems into three classes: not totally uniquely ergodic; totally uniquely ergodic but not topological weakly mixing; totally uniquely ergodic and topological weakly mixing. We will give an affirmative answer to such question for the first two classes. Also, we will show the existence of such dynamical systems in the first class with arbitrary topological entropy.

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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.

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arXiv2023

On multifractal analysis and large deviations of singular-hyperbolic attractors

In this paper we study the multifractal analysis and large derivations for singular hyperbolic attractors, including the geometric Lorenz attractors. For each singular hyperbolic homoclinic class whose periodic orbits are all homoclinically related and such that the space of ergodic probability measures is connected, we prove that: (i) level sets associated to continuous observables are dense in the homoclinic class and satisfy a variational principle; (ii) irregular sets are either empty or are Baire generic and carry full topological entropy. The assumptions are satisfied by $C^1$-generic singular hyperbolic attractors and $C^r$-generic geometric Lorenz attractors $(r\ge 2)$. Finally we prove level-2 large deviations bounds for weak Gibbs measures, which provide a large deviations principle in the special case of Gibbs measures. The main technique we apply is the horseshoe approximation property.

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arXiv2023

On the topological entropy of saturated sets for amenable group actions

Let $(X,ρ,G)$ be a $G-$action topological system, where $G$ is a countable infinite discrete amenable group and $X$ a compact metric space. We prove a variational principle for topological entropy of saturated sets for systems which have specification and uniform separation properties. As an application, we compute the topological entropy of level sets and irregular sets.

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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.

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arXiv2022

Existence and Distributional Chaos of Points that are Recurrent but Not Banach Recurrent

According to the recurrent frequency, many levels of recurrent points are found, such as periodic points, almost periodic points, weakly almost periodic points, quasi-weakly almost periodic points and Banach recurrent points. In this paper, we consider symbolic dynamics and show the existence of six refined levels between Banach recurrence and general recurrence. Despite the fact that these refined levels are all null-measure under any invariant measure, we further show they carry strong topological complexity. Each refined level of recurrent points is dense in the whole space and contains an uncountable distributionally chaotic subset. For a wide range of dynamical systems, such as expansive systems with the shadowing property, we also show the distributional chaos of the points that are recurrent but not Banach recurrent.

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arXiv2021

Uniqueness Of Ergodic Optimization Of Top Lyapunov Exponent For Typical Matrix Cocycles

In this article, we consider the ergodic optimization of the top Lyapunov exponent. We prove that there is a unique maximising measure of top Lyapunov expoent for typical matrix cocyles. By using the results we obtain, we prove that in any non-uniquely ergodic minimal dynamical system, the Lyapunov-irregular points are typical for typical matrix cocyles.

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