arXiv subjects

Changguang Dong

Publications and source records attributed to Changguang Dong.

14 recordsLinked to original sources

arXiv2025

Construction of equilibrium states revisited

In [52], Parmenter and Pollicott establish an abstract criterion that gives a geometric construction of equilibrium states for a class of partially hyperbolic systems. We refine their criterion to cover a much broader class of diffeomorphisms, which include certain diffeomorphisms with exponential mixing property (with respect to volume), Katok maps and ``almost Anosov'' diffeomorphisms. As a special case, we obtain a construction of equilibrium states for ergodic partially hyperbolic affine maps/flows on homogeneous spaces, without any restrictions on the orbit growth along center directions.

math.DS↗

arXiv2025

Slow entropy for abelian actions

We calculate slow entropy type invariant introduced by A. Katok and J.-P. Thouvenot in [5] for higher rank smooth abelian actions for two leading cases: when the invariant measure is absolutely continuous and when it is hyperbolic. We generalize Brin-Katok local entropy Theorem to the abelian action for the above two cases. We also prove that, for abelian actions, the transversal Hausdorff dimensions are universal, i.e. dependent on the action but not on any individual element of the action.

math.DS↗

arXiv2025

On the original Ulam's problem and its quantization

In this paper we show that under general resonance the classical piecewise linear Fermi-Ulam accelerator behaves substantially different from its quantization in the sense that the classical accelerator exhibits typical recurrence and non-escaping while the quantum version enjoys quadratic energy growth in general. We also describe a procedure to locate the escaping orbits, though exceptionally rare in the infinite-volume phase space, for the classical accelerators, which in particular include Ulam's very original proposal and the linearly escaping orbits therein in the existing literature, and hence provide a complete (modulo a null set) answer to Ulam's original question. For the quantum accelerators, we reveal under resonance the direct and explicit connection between the energy growth and the shape of the quasi-energy spectra.

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arXiv2024

On $r$-Neutralized Entropy: Entropy Formula and Existence of Measures Attaining the Supremum

In this article we study $r$-neutralized local entropy and derive some entropy formulas. For an ergodic hyperbolic measure of a smooth system, we show that the $r$-neutralized local entropy equals the Brin-Katok local entropy plus $r$ times the pointwise dimension of the measure. We further establish the existence of ergodic measures that maximize the $r$-neutralized entropy for certain hyperbolic systems. Moreover, we construct a uniformly hyperbolic system, for which such measures are not unique. Finally, we present some rigidity results related to these ergodic measures.

math.DS↗

arXiv2024

Global rigidity of smooth ${\mathbb Z}\ltimes_λ{\mathbb R}$-actions on ${\mathbb T}^2$

For $λ>1$, we consider the locally free ${\mathbb Z}\ltimes_λ{\mathbb R}$ actions on ${\mathbb T}^2$. We show that, if the action is $C^r$ with $r\geq2$, then it is $C^{r-ε}$-conjugate to an affine action generated by a hyperbolic automorphism and a linear translation flow along expanding eigen-direction of the automorphism. In contrast, there exists a $C^{1+α}$-action which is semi-conjugate, but not topologically conjugate to an affine action.

math.DS↗

arXiv2023

Limit theorems for low dimensional generalized $(T,T^{-1})$ transformations

We consider generalized $(T, T^{-1})$ transformations such that the base map satisfies a multiple mixing local limit theorem and anticoncentration large deviation bounds and in the fiber we have $\mathbb{R}^d$ actions with $d=1$ or $2$ which are exponentially mixing of all orders. If the skewing cocycle has zero drift, we show that the ergodic sums satisfy the same limit theorems as the random walks in random scenery studied by Kesten and Spitzer (1979) and Bolthausen (1989). The proofs rely on the quenched CLT for the fiber action and the control of the quenched variance. This paper complements our previous work where the classical central limit theorem is obtained for a large class of generalized $(T, T^{-1})$ transformations.

math.DS↗

arXiv2020

Mixing properties of generalized $T, T^{-1}$ transformations

We study mixing properties of generalized $T, T^{-1}$ transformation. We discuss two mixing mechanisms. In the case the fiber dynamics is mixing, it is sufficient that the driving cocycle is small with small probability. In the case the fiber dynamics is only assumed to be ergodic, one needs to use the shearing properties of the cocycle. Applications include the Central Limit Theorem for sufficiently fast mixing systems and the estimates on deviations of ergodic averages.

math.DS↗

arXiv2020

On the ergodic properties of time changes of partially hyperbolic homogeneous flows

For any accessible partially hyperbolic homogeneous flow, we show that all smooth time changes are K and hence mixing of all orders. We also establish stable ergodicity for time-one map of these time changes.

math.DS↗

arXiv2020

Rigidity of joinings for some measure preserving systems

We introduce two properties: strong R-property and $C(q)$-property, describing a special way of divergence of nearby trajectories for an abstract measure preserving system. We show that systems satisfying the strong R-property are disjoint (in the sense of Furstenberg) with systems satisfying the $C(q)$-property. Moreover, we show that if $u_t$ is a unipotent flow on $G/Γ$ with $Γ$ irreducible, then $u_t$ satisfies the $C(q)$-property provided that $u_t$ is not of the form $h_t\times\operatorname{id}$, where $h_t$ is the classical horocycle flow. Finally, we show that the strong R-property holds for all (smooth) time changes of horocycle flows and non-trivial time changes of bounded type Heisenberg nilflows.

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arXiv2020

Flexibility of statistical properties for smooth systems satisfying the central limit theorem

In this paper we exhibit new classes of smooth systems which satisfy the Central Limit Theorem (CLT) and have (at least) one of the following properties: (1) zero entropy; (2) weak but not strong mixing; (3) (polynomially) mixing but not $K$; (4) $K$ but not Bernoulli; (5) non Bernoulli and mixing at arbitrary fast polynomial rate. We also give an example of a system satisfying the CLT where the normalizing sequence is regularly varying with index $1$.

math.DS↗

arXiv2019

Bernoulli property for certain skew products over hyperbolic systems

We study the Bernoulli property for a class of partially hyperbolic systems arising from skew products. More precisely, we consider a hyperbolic map $(T,M,μ)$, where $μ$ is a Gibbs measure, an aperiodic Hölder continuous cocycle $ϕ:M\to \mathbb R$ with zero mean and a zero-entropy flow $(K_t,N,ν)$. We then study the skew product $$ T_ϕ(x,y)=(Tx,K_{ϕ(x)}y), $$ acting on $(M\times N,μ\times ν)$. We show that if $(K_t)$ is of slow growth and has good equidistribution properties, then $T_ϕ$ remains Bernoulli. In particular, our main result applies to $(K_t)$ being a typical translation flow on a surface of genus $\geq 1$ or a smooth reparametrization of isometric flows on $\mathbb T^2$. This provides examples of non-algebraic, partially hyperbolic systems which are Bernoulli and for which the center is non-isometric (in fact might be weakly mixing).

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arXiv2018

Rigidity of a class of smooth singular flows on $\mathbb T^2$

We study joining rigidity in the class of von Neumann flows with one singularity. They are given by a smooth vector field $\mathcal{X}$ on $\mathbb T^2\setminus \{a\}$, where $\mathcal{X}$ is not defined at $a\in \mathbb T^2$. It follows that the phase space can be decomposed into a (topological disc) $D_\mathcal{X}$ and an ergodic component $E_\mathcal{X}=\mathbb T^2\setminus D_\mathcal{X}$. Let $ω_\mathcal{X}$ be the 1-form associated to $\mathcal{X}$. We show that if $|\int_{E_{\mathcal{X}_1}}dω_{\mathcal{X}_1}|\neq |\int_{E_{\mathcal{X}_2}}dω_{\mathcal{X}_2}|$, then the corresponding flows $(v_t^{\mathcal{X}_1})$ and $(v_t^{\mathcal{X}_2})$ are disjoint. It also follows that for every $\mathcal{X}$ there is a uniquely associated frequency $α=α_{\mathcal{X}}\in \mathbb T$. We show that for a full measure set of $α\in \mathbb T$ the class of smooth time changes of $(v_t^\mathcal{X_α})$ is joining rigid, i.e. every two smooth time changes are either cohomologous or disjoint. This gives a natural class of flows for which the answer to a problem of Ratner (Problem 3 in \cite{Rat4}) is positive.

math.DS↗

arXiv2017

On density of infinite subsets I

Let $Y$ be a compact metric space, $G$ be a group acting by transformations on $Y$. For any infinite subset $A\subset Y$, we study the density of $gA$ for $g\in G$ and quantitative density of the set $\displaystyle{\bigcup_{g\in G_n}gA}$ by the Hausdorff semimetric $d^H$. It is proven that for any integer $n\ge 2$, $ε>0$, any infinite subset $A\subset \mathbb T^n$, there is a $g\in SL(n,\mathbb Z)$ such that $gA$ is $ε$-dense. We also show that, for any infinite subset $A\subset [0,1]$, for generic rotation and generic 3-IET, $$\liminf_nn\cdot d^H\left(\bigcup_{k=0}^{n-1}T^kA,[0,1]\right)=0.$$

math.DS↗

arXiv2017

On density of infinite subsets II: dynamics on homogeneous spaces

Let $G$ be a noncompact semisimple Lie group, $Γ$ be an irreducible cocompact lattice in $G$, and $P 0$, there is a $g\in P$ such that $gA$ is $ε$-dense. We also prove a similar result for certain discrete group actions on $\mathbb T^n$.

math.DS↗