13 problems

Sinai's conjecture. For sufficiently large $k$, the standard map $T$ has positive metric entropy with respect to Lebesgue measure.

Standard map entropy conjecture. The map $f_{\lambda}$ has positive metric entropy for every $\lambda\neq 0$. Despite its simple formulation and extensive study, this conjecture re…

Let $\varepsilon_{\mathrm{c}}(\omega)$ be the critical function for an irrational rotation number $\omega$, and let $\mathcal{R}_{k}(\varepsilon)$ denote the residue of the perturb…

Let $f_k:\mathbb{T}^2\to\mathbb{T}^2$ be the standard map … A $C^2$-diffeomorphism $g$ is sufficiently $C^2$-close to $f_k$ when it lies in a sufficiently small $C^2$ neighborhood…

Let the standard map be the one-parameter family of area-preserving analytic diffeomorphisms of the two-dimensional torus $\mathbb T^2$. Its phase space is expected to exhibit both…

Let $q$ range over even denominators, and consider the periodic trajectories of the transformation $f$. Large-period distribution conjecture. The distributions of large periods of…

Let $q$ be the denominator of the rational parameter in the transformation $f$, and consider its bounded trajectories. Bounded-trajectory length conjecture. The maximum length of t…

Let $b1=1/q$ and $y_0=0$, and let $ell(q)$ denote the unique odd-length period under $f$ on $L_o$. Quadratic escaping-orbit length conjecture. The length of the escaping orbit grow…

Let $b1=p/q$ be rational, and let $y_0=a/b$ be an initial level for the original map. An orbit is escaping if it is unbounded. Escaping-orbit existence conjecture. For every…

Greene's golden-mean conjecture. The rotation number of the critical invariant curve is equal to the golden mean:

Let $f_k\blankcolon\blankmathbb{T}^2\to\mathbb{T}^2$ be the standard family … where $k>0$. Sinai's conjecture. For many values of the parameter $k>0$, the map $f_k$ has positive me…

Let $f_\kappa$ denote the Standard Map with coupling constant $\kappa$, and consider the set of parameter values for which $f_\kappa$ has at least one elliptic island. Carleson's c…

Let $f_k:\mathbb{T}^2\to\mathbb{T}^2$ be the Taylor–Chirikov standard map, and let $A$ be the set of parameters $k$ for which $f_k$ is ergodic and non-uniformly hyperbolic with res…