Huichi Huang

Huichi Huang
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Huichi Huang
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Mathematics - Dynamical Systems (6)
 
Mathematics - Operator Algebras (5)
 
Mathematics - Quantum Algebra (2)
 
Mathematics - Classical Analysis and ODEs (2)
 
Mathematics - Number Theory (1)
 
Mathematics - Functional Analysis (1)

Publications Authored By Huichi Huang

Given a sequence $\Sigma=\{F_n\}$ of finite subsets in a discrete group $\Gamma$, suppose $b$ in $\Gamma$ is fixed by $\Sigma$ from right. If a subset $\Lambda$ of $\Gamma$ has positive upper density with respect to $\Sigma$, then for any positive integer $k$, there exist $n>0$ and $a\in\Gamma$ such that $\{b^{jn}a\}_{j=0}^{k-1}\subseteq\Lambda$. Read More

We generalize the concept of stabilizer subgroups to compact quantum groups. Read More

We express continuous $\times p,\times q$-invariant measures on the unit circle via some simple forms. On one hand, a continuous $\times p,\times q$-invariant measure is the weak-$*$ limit of average of Dirac measures along an irrational orbit. On the other hand, a continuous $\times p,\times q$-invariant measure is a continuous function on $[0,1]$ satisfying certain function equations. Read More

We define densities $D_\Sigma(A)$, $\overline{D}_\Sigma(A)$ and $\underline{D}_\Sigma(A)$ for a subset $A$ of $\mathbb{N}$ with respect to a sequence $\Sigma$ of finite subsets of $\mathbb{N}$ and study Fourier coefficients of ergodic, weak-mixing and strong-mixing $\times p$-invariant measures on the unit circle $\mathbb{T}$. Combining these, we prove the following measure rigidity results: on $\mathbb{T}$, the Lebesgue measure is the only continuous $\times p$-invariant measure satisfying one of the following: (1) $\mu$ is ergodic and there exist a F{\o}lner sequence $\Sigma$ in $\mathbb{N}$ and a nonzero integer $l$ such that $\mu$ is $\times (p^j+l)$-invariant for all $j$ in a subset $A$ of $\mathbb{N}$ with $D_\Sigma(A)=1$, (2) $\mu$ is weak-mixing and there exist a F{\o}lner sequence $\Sigma$ in $\mathbb{N}$ and a nonzero integer $l$ such that $\mu$ is $\times (p^j+l)$-invariant for all $j$ in a subset $A$ of $\mathbb{N}$ with $\overline{D}_\Sigma(A)>0$, (3) $\mu$ is strong-mixing and there exists a nonzero integer $l$ such that $\mu$ is $\times (p^j+l)$-invariant for infinitely many $j$. Moreover, a $\times p$-invariant measure satisfying (2) or (3) is either the Dirac measure $\delta_0$ or the Lebesgue measure. Read More

We prove a mean ergodic theorem for amenable discrete quantum groups. As an application, we prove a Wiener type theorem for continuous measures on compact metrizable groups. Read More

Motivated by reformulating Furstenberg's $\times p,\times q$ conjecture via representations of a crossed product $C^*$-algebra, we show that in a discrete $C^*$-dynamical system $(A,\Gamma)$, the space of (ergodic) $\Gamma$-invariant states on $A$ is homeomorphic to a subspace of (pure) state space of $A\rtimes\Gamma$. Various applications of this in topological dynamical systems and representation theory are obtained. In particular, we prove that the classification of ergodic $\Gamma$-invariant regular Borel probability measures on a compact Hausdorff space $X$ is equivalent to the classification a special type of irreducible representations of $C(X)\rtimes \Gamma$. Read More

We show that equidistribution of irrational orbits on the unit circle implies Furstenberg's conjecture. Read More

We investigate compact quantum group actions on unital $C^*$-algebras by analyzing invariant subsets and invariant states. In particular, we come up with the concept of compact quantum group orbits and use it to show that countable compact metrizable spaces with infinitely many points are not quantum homogeneous spaces. Read More

We construct faithful actions of quantum permutation groups on connected compact metrizable spaces. This disproves a conjecture of Goswami. Read More