arXiv subjects

Qiujie Qiao

Publications and source records attributed to Qiujie Qiao.

3 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↗

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↗