arXiv subjects

Daren Wei

Publications and source records attributed to Daren Wei.

12 recordsLinked to original sources

arXiv2025

Time change rigidity for unipotent flows

We prove a dichotomy regarding the behavior of one-parameter unipotent flows on quotients of semisimple lie groups under time change. We show that if $u^{(1)}_t$ acting on $\mathbf{G}_{1}/Γ_1$ is such a flow it satisfies exactly one of the following: (1) The flow is loosely Kronecker, and hence measurably isomorphic after an appropriate time change to any other loosely Kronecker system. (2) The flow exhibits the following rigid behavior: if the one-parameter unipotent flow $u^{(1)} _ t$ on $\mathbf{G}_1/Γ_1$ is measurably isomorphic after time change to another such flow $u^{(2)} _ t$ on $\mathbf{G}_2/Γ_ 2$, then $\mathbf{G}_1/Γ_1 $ is isomorphic to $\mathbf{G}_2/ Γ_2$ with the isomorphism taking $u^{(1)}_t$ to $u^{(2)}_t$ and moreover the time change is cohomologous to a trivial one up to a renormalization.

math.DS↗

arXiv2023

Time change for unipotent flows and rigidity

We prove a dichotomy regarding the behavior of one-parameter unipotent flows on quotients of semisimple lie groups under time change. We show that if $u^{(1)}_t$ acting on $G_{1}/Γ_1$ is such a flow it satisfies exactly one of the following: (1) The flow is loosely Kronecker, and hence isomorphic after an appropriate time change to any other loosely Kronecker system. (2) The flow exhibits the following rigid behavior: if the one-parameter unipotent flow $u^{(1)} _ t$ on $G_1/Γ_1$ is isomorphic after time change to another such flow $u^{(2)} _ t$ on $G_2/Γ_ 2$, then $G_1/Γ_1 $ is isomorphic to $G_2/ Γ_2$ with the isomorphism taking $u^{(1)} _ t$ to $u^{(2)} _ t$ and moreover the time change is cohomologous to a trivial one. The full details will appear in [LW23].

math.DS↗

arXiv2021

Slow entropy of some combinatorial constructions

Measure-theoretic slow entropy is a more refined invariant than the classical measure-theoretic entropy to characterize the complexity of dynamical systems with subexponential growth rates of distinguishable orbit types. In this paper we prove flexibility results for the values of upper and lower polynomial slow entropy of rigid transformations as well as maps admitting a good cyclic approximation. Moreover, we show that there cannot exist a general upper bound on the lower measure-theoretic slow entropy for systems of finite rank.

math.DS↗

arXiv2021

Slow entropy for some Anosov-Katok diffeomorphisms

The Anosov-Katok method is one of the most powerful tools of constructing smooth volume-preserving diffeomorphisms of entropy zero with prescribed ergodic or topological properties. To measure the complexity of systems with entropy zero, invariants like slow entropy have been introduced. In this article we develop several mechanisms facilitating computation of topological and measure-theoretic slow entropy of Anosov-Katok diffeomorphisms.

math.DS↗

arXiv2021

Kakutani equivalence for products of some special flows over rotations

We study Kakutani equivalence for products of some special flows over rotations with roof function smooth except a singularity at $0\in\mathbb{T}$. We estimate the Kakutani invariant for product of these flows with different powers of singularities and rotations from a full measure set. As a corollary, we obtain a countable family of pairwise non-Kakutani equivalent products of special flows over rotations.

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.

math.DS↗

arXiv2020

Slow entropy of higher rank abelian unipotent actions

We study slow entropy invariants for abelian unipotent actions $U$ on any finite volume homogeneous space $G/Γ$. For every such action we show that the topological slow entropy can be computed directly from the dimension of a special decomposition of $\operatorname{Lie}(G)$ induced by $\operatorname{Lie}(U)$. Moreover, we are able to show that the metric slow entropy of the action coincides with its topological slow entropy. As a corollary, we obtain that the complexity of any abelian horocyclic action is only related to the dimension of $G$. This generalizes the rank one results from [A. Kanigowski, K. Vinhage, D. Wei, Commun. Math. Phys. 370 (2019), no. 2, 449-474.] to higher rank abelian actions.

math.DS↗

arXiv2020

Survey on entropy-type invariants of sub-exponential growth in dynamical systems

Measure-theoretic and topological entropy are classical invariants in the theory of dynamical systems. There are several recently developed entropy type invariants for systems of sub-exponential growth: sequence entropy, slow entropy, Kakutani invariants, scaled entropy, entropy dimensions and entropy convergence rate. They measure the complexity of zero entropy systems by different approaches. These new invariants and corresponding new theories have many applications and interesting properties. This survey paper gives a comprehensive exposition of the slow entropy theory and also discusses some related topics.

math.DS↗

arXiv2019

Kakutani Equivalence of Unipotent Flows

We study Kakutani equivalence in the class of unipotent flows acting on finite volume quotients of semisimple Lie groups. For every such flow we compute the Kakutani invariant of M. Ratner, the value of which being explicitly given by the Jordan block structure of the unipotent element generating the flow. This in particular answers a question of M. Ratner. Moreover it follows that the only standard unipotent flows are given by $\begin{pmatrix} 1 & t \\ 0 & 1 \end{pmatrix} \times \operatorname{id}$ acting on $(SL(2,\mathbb{R}) \times G')/Γ'$, where $Γ'$ is an irreducible lattice in $SL(2,\mathbb{R}) \times G'$ (with the possibility that $G' = \{e\}$).

math.DS↗

arXiv2018

Horocycle flow on negative variable curvature surface is standard

We provide a new proof that the horocycle flow preserving the Margulis measure on a variable negative curvature surface is standard. This was first proved by Ratner. The main purpose of this note is to provide a simplified case of the arguments for Kakutani equivalence of unipotent flows on homogeneous spaces, which have similar but more complicated structures, as well as illustrate the versatility of the method by applying it to a non-homogeneous flow.

math.DS↗

arXiv2017

Slow Entropy of Some Parabolic Flows

We study nontrivial entropy invariants in the class of parabolic flows on homogeneous spaces, quasi-unipotent flows. We show that topological complexity (ie, slow entropy) can be computed directly from the Jordan block structure of the adjoint representation. Moreover using uniform polynomial shearing we are able to show that the metric orbit growth (ie, slow entropy) coincides with the topological one, establishing hence variational principle for quasi-unipotent flows (this also applies to the non-compact case). Our results also apply to sequence entropy. We establish criterion for a system to have trivial topological complexity and give some examples in which the measure-theoretic and topological complexities do not coincide for uniquely ergodic systems, violating the intuition of the classical variational principle.

math.DS↗

arXiv2017

Product of two Kochergin flows with different exponents is not standard

We study the standard(zero entropy loosely Bernoulli or loosely Kronecker) property for products of Kochergin smooth flows on $\mathbb{T}^2$ with one singularity. These flows can be represented as special flows over irrational rotations of the circle and under roof functions which are smooth on $\mathbb{T}^2\setminus \{0\}$ with a singularity at $0$. We show that there exists a full measure set $\mathscr{D}\subset\mathbb{T}$ such that the product system of two Kochergin flows with different power of singularities and rotations from $\mathscr{D}$ is not standard.

math.DS↗