17 problems

Fried's conjecture. The function $\zeta_{g,\rho}(s)$ is well-defined and nonvanishing at $s=0$, and

Let $f$ be a hyperbolic rational map, let $Z(s)$ denote its dynamical zeta function, let \\{\mu_j\} be the zeros of $Z$ counted with multiplicity, and let $\delta$ be the dimensi…

Conjecture B. The SRB measure $\mu$ has exponential rates of mixing if and only if $\Lambda_j$ is mixing, and this holds if and only if $z=1$ is the only zero of modulus one of…

Conjecture B. For $\omega\ne 0$, if $\omega$ belongs to the analytic correlation spectrum, then either $d_{f|_{\Lambda_j}}(z)$ is not holomorphic at $z=\exp(i\omega)$ or it vanishe…

Let $f$ be an analytic surface diffeomorphism with dominated splitting, let ${\Cal N}$ denote the relevant set of periodic points, let $\lambda_F$ be the associated quantity on per…

Resonance-multiplicity conjecture. The following assertions hold:

Generic semisimplicity conjecture. For the trivial representation $\rho=\rho_{\mathrm{triv}}$, the condition

Non-entire continuation conjecture. The function $\eta_D$ cannot be prolonged as an entire function.

Essential spectral gap conjecture. For every $\omega\in\mathcal{H}^1(X)$,

Generic-domain frequency-separation conjecture. For generic domains there exists a sequence $\{\lambda_{m_j}\}$ satisfying this separation condition. If such a sequence exists unde…

Let $\eta_D$ be the Dirichlet dynamical zeta function associated with the oriented periodic rays of a dispersing billiard flow, initially defined by its convergent Dirichlet series…

Extension conjecture. The method should be improvable so as to obtain a complete understanding of the extension of the dynamical zeta function in the differentiable context.

Let $K$ be an algebraically closed field of characteristic $0$, let $\phi\in K(z)$ be a quadratic polynomial, and for $m>0$ define its dynamical zeta function by … Here…

Let $(\Sigma,g)$ be a negatively curved oriented compact Riemannian manifold of dimension $n>2$, and let $\alpha$ be an acyclic unitary representation of $\pi_1(\Sigma)$. Let…

Holomorphic extension and zero localization conjecture. The semiclassical zeta function $Z_{sc}(s)$ for contact Anosov flows has a holomorphic extension to the whole complex plane…

Error-term conjecture. The above bound holds for all contact Anosov flows. The authors state that they cannot prove this in full generality and explain that their result shows the…

Let ${\mathcal P}$ be the set of primitive periodic rays. For each $\gamma\in{\mathcal P}$, define … and let $r_\gamma=0$ if $m_\gamma$ is even and $r_\gamma=1$ if $m_\gamma$ is od…