arXiv subjects
Publications and source records attributed to Xueting Tian.
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.
In this article, we pay attention to transitive dynamical systems having the shadowing property and the entropy functions are upper semicontinuous. As for these dynamical systems, when we consider ergodic optimization restricted on the subset of invariant measures whose metric entropy are equal or greater than a given constant, we prove that for generic real continuous functions the ergodic optimization measure is unique, ergodic, full support and have metric entropy equal to the given constant. Similar results also hold for suspension flows over transitive subshift of finite type, Cr (r\geq 2)- generic geometric Lorenz attractors and C1-generic singular hyperbolic attractors.
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.
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.
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.
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.
For $C^0$ generic continuous maps or homeomorphisms on compact Riemannian manifold, we prove that (1) the space of physical-like measures coincides with the set of invariant measures supported on chain recurrent classes, (2) every point in the base space is typical (that is, for any point in the base space, its empirical measures are contained in the space of physical-like measures) and (3) there is a subset of strongly regular set with Lebesgue zero measure but has infinite topological entropy. Moreover, some comparison between $C^0$ generic systems and $C^0$ conservative generic systems are discussed.
It follows from Oseledec Multiplicative Ergodic Theorem (or Kingmans Subadditional Ergodic Theorem) that the Lyapunov-irregular set of points for which the Oseledec averages of a given continuous cocycle diverge has zero measure with respect to any invariant probability measure. In strong contrast, for any dynamical system f with exponential specification property and a Holder continuous matrix cocycle A, we show here that if there exist ergodic measures with different Lyapunov spectrum, then the Lyapunov-irregular set of A displays distributional chaos of type 1.
In this paper we define unstable topological entropy for any subsets (not necessarily compact or invariant) in partially hyperbolic systems as a Carathéodory dimension characteristic, motivated by the work of Bowen and Pesin etc. We then establish some basic results in dimension theory for Bowen unstable topological entropy, including an entropy distribution principle and a variational principle in general setting. As applications of this new concept, we study unstable topological entropy of saturated sets and extend some results in \cite{Bo, PS2007}. Our results give new insights to the multifractal analysis for partially hyperbolic systems.
There are lots of results to study dynamical complexity on irregular sets and level sets of ergodic average from the perspective of density in base space, Hausdorff dimension, Lebesgue positive measure, positive or full topological entropy (and topological pressure) etc.. However, it is unknown from the viewpoint of chaos. There are lots of results on the relationship of positive topological entropy and various chaos but it is known that positive topological entropy does not imply a strong version of chaos called DC1 so that it is non-trivial to study DC1 on irregular sets and level sets. In this paper we will show that for dynamical system with specification property, there exist uncountable DC1-scrambled subsets in irregular sets and level sets. On the other hand, we also prove that several recurrent levels of points with different recurrent frequency all have uncountable DC1-scrambled subsets. The main technique established to prove above results is that there exists uncountable DC1-scrambled subset in saturated sets.
In [30] different statistical behavior of dynamical orbits without syndetic center are considered. In present paper we continue this project and consider different statistical behavior of dynamical orbits with nonempty syndetic center: Two of sixteen cases appear (for which other fourteen cases are still unknown) in transive topologically expanding or hyperbolic systems and are discovered to have full topological entropy for which it is also true if combined with non-recurrence and multifractal analysis such as quasi-regular set, irregular set and level sets. In this process a strong entropy-dense property, called minimal-entropy-dense, is established. In particular, we show that points that are minimal (or called almost periodic), a classical and important concept in the study of dynamical systems, form a set with full topological entropy if the dynamical system satisfies shadowing or almost specification property.
The topological entropy of various gap-sets on periodic-like recurrence and Birkhoff regularity were considered in [69] but some Banach recurrence and Lyapunov regularity are not considered. In this paper we introduce five new levels on Banach recurrence and show they all carry full topological entropy, and simultaneously combine with Lyapunov regularity to get some refined theory on mixed multifractal analysis of [8,29]. In this process, we strengthen Pfister and Sullivan's result of [58] from saturated property to transitively-saturated property (and from single-saturated property to transitively-convex-saturated property).
In this paper we mainly study the dynamical complexity of Birkhoff ergodic average under the simultaneous observation of any number of continuous functions. These results can be as generalizations of [6,35] etc. to study Birkhorff ergodic average from one (or finite) observable function to any number of observable functions from the dimensional perspective. For any topological dynamical system with $g-$almost product property and uniform separation property, we show that any {\it jointly-irregular set}(i.e., the intersection of a series of $ϕ-$irregular sets over several continuous functions) either is empty or carries full topological pressure. In particular, if further the system is not uniquely ergodic, then the {\it completely-irregular set}(i.e., intersection of all possible {\it nonempty $ϕ-$irregular} sets) is nonempty(even forms a dense $G_δ$ set) and carries full topological pressure. Moreover, {\it irregular-mix-regular sets} (i.e., intersection of some $ ϕ-$irregular sets and $ φ-$regular sets) are discussed. Similarly, the above results are suitable for the case of BS-dimension. As consequences, these results are suitable for any system such as shifts of finite type or uniformly hyperbolic diffeomorphisms, time-1 map of uniformly hyperbolic flows, repellers, $β-$shifts etc..
We prove that when $f$ is a continuous selfmap acting on compact metric space $(X,d)$ which satisfies the shadowing property, then the set of irregular points (i.e. points with divergent Birkhoff averages) has full entropy. Using this fact we prove that in the class of $C^0$-generic maps on manifolds, we can only observe (in the sense of Lebesgue measure) points with convergant Birkhoff averages. In particular, the time average of atomic measures along orbit of such points converges to some SRB-like measure in the weak$^*$ topology. Moreover, such points carry zero entropy. In contrast, irregular points are non-observable but carry infinite entropy.
In this paper we prove that for an ergodic hyperbolic measure $ω$ of a $C^{1+α}$ diffeomorphism $f$ on a Riemannian manifold $M$, there is an $ω$-full measured set $\widetildeΛ$ such that for every invariant probability $μ\in \mathcal{M}_{inv}(\widetildeΛ,f)$, the metric entropy of $μ$ is equal to the topological entropy of saturated set $G_μ$ consisting of generic points of $μ$: $$h_μ(f)=h_{\top}(f,G_μ).$$ Moreover, for every nonempty, compact and connected subset $K$ of $\mathcal{M}_{inv}(\widetildeΛ,f)$ with the same hyperbolic rate, we compute the topological entropy of saturated set $G_K$ of $K$ by the following equality: $$\inf\{h_μ(f)\mid μ\in K\}=h_{\top}(f,G_K).$$ In particular these results can be applied (i) to the nonuniformy hyperbolic diffeomorphisms described by Katok, (ii) to the robustly transitive partially hyperbolic diffeomorphisms described by ~Ma{ñ}{é}, (iii) to the robustly transitive non-partially hyperbolic diffeomorphisms described by Bonatti-Viana. In all these cases $\mathcal{M}_{inv}(\widetildeΛ,f)$ contains an open subset of $\mathcal{M}_{erg}(M,f)$.
We study a class of asymptotically entropy-expansive $C^1$ diffeomorphisms with dominated splitting on a compact manifold $M$, that satisfy the specification property. This class includes, in particular, transitive Anosov diffeomorphisms and time-one maps of transitive Anosov flows. We consider the nonempty set of physical-like measures that attracts the empirical probabilities (i.e. the time averages) of Lebesgue-almost all the orbits. We define the set $I_f \cap Γ_f \subset M$ of irregular points without physical-like behaviour. We prove that, if not all the invariant measures of $f$ satisfy Pesin Entropy Formula (for instance in the Anosov case), then $I_f \cap Γ_f$ has full topological entropy. We also obtain this result for some class of asymptotically entropy-expansive continuous maps on a compact metric space, if the set of physical-like measures are equilibrium states with respect to some continuous potential. Finally, we prove that also the set $(M \setminus I_f) \cap Γ_f$ of regular points without physical-like behaviour, has full topological entropy.
In this paper we consider a non-atomic invariant hyperbolic measure $μ$ of a $C^1$ diffeomorphsim on a compact manifold, in whose Oseledec splitting the stable bundle dominates the unstable bundle on $μ$ a.e. points. We show an \textit{exponentially} shadowing and an \textit{exponentially} closing lemma, and as applications we show two classical results. One is that there exists a hyperbolic periodic point such that the closure of its unstable manifold has \textit{positive} measure and it has a homoclinic point from which one can deduce a horseshoe. Moreover, such hyperbolic periodic points are dense in the support $supp(μ)$ of the given hyperbolic measure. Another is to show Livshitz Theorem.
In this paper, we explore a topological system $f:M\rightarrow M$ with average shadowing property. We extend Sigmund's results and show that every non-empty, compact and connected subset $V\subseteq\mathcal {M}_{inv}(f)$ coincides with $V_f(y)$, where $\mathcal {M}_{inv}(f)$ denotes the space of invariant Borel probability measures on M, and $V_f(y)$ denotes the accumulation set of time average of Dirac measures supported at the orbit of $y$. We also show that the set $M_{V}=\{y\in M\,\,|\,\,V_{f}(y)=V\}$ is dense in $Δ_{V}=\bigcup_{ν\in V}supp(ν)$. In particular, if $Δ_{max}=\bigcup_{ν\in\mathcal {M}_{inv}(f)}supp(ν)$ is isolated or coincides with $M$, then $M_{max}=\{y: V_{f}(y)=\mathcal {M}_{inv}(f)\}$ is residual in $Δ_{max}$.