Relative ($\tau$) is equivalent to a statement that the sequence of Cayley graphs associated to group quotients $\Gamma_q=\Gamma/N_q$, $q\in\mathbb{N}$, form an expander family. There is a philosophy that expander graphs give rise to good mixing; for instance, one has exponential mixing for the geodesic flow uniformly the along a family of congruence covers of the modular surface, stemming from the symmetry in the $\mathrm{SL}(2,\mathbb{R})$ action and the uniform spectral gap for the Laplacian. Do we see similar phenomena in less structured settings? We investigate this question for a tower of finite sheeted covers of certain expanding dynamical systems. We use transfer operator machinery that applies in particular in the cases of subshifts of finite type and expanding interval maps. We make fruitful connections with KMS states of Cuntz--Krieger algebras.
We present an example that shows that the double Hilbert transform $\H$ can have the characteristic function of an open set of finite measure in $\mathbb{R} \times \mathbb{R}$ as an approximate eigenvector for the eigenvalue $1$. The same holds for the dyadic model of $\H$, the double dyadic shift. We further construct...
E. Abakumov, K. Domelevo, S. Petermichl et al.· 0 citations
We prove that a lattice $\Gamma$ in $\mathrm{PSL}_2(\mathbb{R})$ or $\mathrm{PSL}_2(\mathbb C)$ has positive trace gap, meaning that its traces are uniformly separated, if and only if it is derived from an admissible quaternion algebra. For cocompact Fuchsian groups, this proves the positive trace gap conjecture attrib...
We develop a perturbative KAM reducibility theory for one-frequency $\mathrm{SL}(2,\mathbb{R})$ cocycles based on an admissible Fourier length $\ell$. The regularity relevant to the iteration is measured by positive adapted Fourier width rather than ordinary smoothness in the Euclidean length $|n|$. The same length gov...
Let $A$ be an $\alpha$-H\"older cocycle over a transitive two-sided subshift of finite type, and assume that $A$ is strongly bunched. We prove that if there exist $k\in\{1,\ldots,d-1\}$ and $c>0$ such that \[ \lambda_k(p)-\lambda_{k+1}(p)\ge c \] for every periodic point $p$, then $A$ admits a dominated splitting of in...
We show that if $G$ is a real semisimple Lie group and $\Gamma<G$ is a discrete subgroup with slow growth, in the sense that its growth indicator function is smaller than $\rho$, then $\Gamma$ acts totally dissipatively on the Furstenberg boundary of $G$. Moreover, for $G= SO(n,2)$ and $n\geq 3$, we construct infinite-...
S. Dey, Mikołaj Frączyk, Sebastian Hurtado· 1 citation
Han, Jouhet and Zeng established $q$-analogues of the $\gamma$-expansion formulas for Eulerian polynomials of types $A$ and $B$. Combinatorial interpretations of the corresponding coefficients$a_{n,k}(q)$ and $b_{n,k}(q)$, however, remained open. In this paper, we provide combinatorial interpretations for thesecoeffici...
Tai-Feng Ding, Lin-Tong Wang, Sherry H. F. Yan· 0 citations
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