147 problems

Let $f$ be a smooth map and suppose that all its Lyapunov exponents are nonzero at Lebesgue almost every point. Viana's conjecture. The map $f$ admits an SRB measure. This conjectu…

Let $G_t^{KZ}$ be the Kontsevich–Zorich cocycle over the Teichmüller flow, and let $\mu_\kappa^{(1)}$ be its canonical absolutely continuous invariant measure. Zorich–Kontsevich co…

Continuity conjecture. The Lyapunov exponents vary continuously as functions of $f$ in the space of $\mathcal{C}^2$ volume-preserving Anosov diffeomorphisms on $\mathbb T^3$.

Lyapunov-exponent conjecture. The associated Lyapunov exponents are closely related to the parabolic degrees of $\mathscr{E}^{3,0}$ and $\mathscr{E}^{2,1}$.

A Shimura–Teichmüller curve is a Teichmüller curve, equivalently a Veech surface orbit, with totally degenerate Lyapunov spectrum. Möller's conjecture. There are no Shimura–Teichmü…

Zorich conjecture. The Lyapunov spectrum of the Zorich cocycle is simple; equivalently, all the $\theta_i$ are distinct.

Eden conjecture. The maximum in the formula for $d_L$ is attained on an equilibrium or periodic orbit, rather than on a chaotic orbit. The conjecture concerns when the global Lyapu…

Kontsevich–Zorich conjecture. For the hyperelliptic components $\mathcal{M}^{\mathcal{H}}_{2g-2}$ and $\mathcal{M}^{\mathcal{H}}_{(g-1,g-1)}$,

Positive Lyapunov exponent conjecture. One has $\Lambda>0$.

Let $H_{b}^{\alpha}(X,GL(2,\mathbb R))$ denote the class of the cocycles considered in the source. Viana's continuity conjecture. Lyapunov exponents are always continuous among…

Let $M$ be a smooth compact manifold and let $m$ be a smooth volume measure. For a generic conservative diffeomorphism $f$, consider its Lyapunov exponents and Oseledets splitting.…

Let $P_N$ be a random matrix product, and write its eigenvalues as … The quantities $\lambda_k$ are the stability exponents, while the Lyapunov exponents are defined from the eigen…

Hammel–Yorke–Grebogi's conjecture. A typical dissipative map $f:\mathds{R}^2\to\mathds{R}^2$ with positive Lyapunov exponent satisfies $\operatorname{FinHolSh}(1/2,1/2)$. This conj…

Kontsevich–Zorich conjecture. The Lyapunov exponents are (a) all non-zero, expressing non-uniform hyperbolicity, and (b) all distinct, expressing simplicity.

Zorich's conjectures. If $\nu_g>0$, then there exists a Lagrangian subspace $L_g$ such that the cycles remain at bounded distance from $L_g$. If, moreover, the exponents $\nu_j$, f…

Let $\theta_1,\dots,\theta_g$ be the Lyapunov exponents of the Rauzy–Veech cocycle, ordered so that … A complete Lagrangian flag is a nested sequence of Lagrangian subspaces whose…

Let $\theta_1>\theta_2>\dots>\theta_g$ be the Lyapunov exponents of the Rauzy–Veech cocycle. The asymptotic deviation theorem uses these exponents to determine the dimensions of th…

Let $S$ be a generic flat surface in a connected component of a stratum, and let $\nu_1,\dots,\nu_g$ be the Lyapunov exponents associated with the Teichmüller geodesic flow. The as…

Let $f$ be a non-uniformly expanding map. An orbit has a zero Lyapunov exponent if its Lyapunov exponent is zero, and $f$ has exponential decay of correlations if its correlations…

Let $f$ and $g$ be topologically conjugate one-dimensional maps, and let $x$ be a point of $f$ that is not attracted to a periodic orbit. The upper pointwise Lyapunov exponent of…

Let $f$ be a dynamical system and let $\tau(B(x,r))$ denote the first return time of the ball $B(x,r)$ to itself. Let $\mu$ be an ergodic hyperbolic measure with nonzero entropy…

All-energy localization conjecture. As for $d=1$, localization holds at all energies and for arbitrary disorder in dimension $d=2$.

Let ${\cal M}^{\cal H}_{(2g-2)}$ and ${\cal M}^{\cal H}_{(g-1,g-1)}$ denote the hyperelliptic components of the corresponding moduli spaces of holomorphic differentials, and let…

Fix integers ${\bf d}=(d_1,\dots,d_n)$ with each $d_i$ positive or equal to $-1$, satisfying $\sum_i d_i=4g-4$, and let ${\cal Q}_{\bf d}$ be the moduli space of non-square meromor…

Let $E_c$ be the energy threshold, $N$ the system size, and $d$ the dimension parameter. Let ${\cal T}$ be the dynamics, $\mu$ its invariant measure, and $P_i^k$ and $P_j^l$ distin…