56 problems

Consider the random dynamical system induced by the stochastic differential equation, and fix $a>0$ and $\beta\in\mathbb{R}$. Let $\Lambda_1$ denote its largest Lyapunov exponent.…

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

Let $G=(V,E,l)$ be a connected metric graph endowed with an interval exchange transformation $T$ partitioning $G$ into $n$ intervals with lengths $\ell_1, \ell_2, \ldots, \ell_n$ l…

Periodic-orbit Lyapunov rigidity conjecture. If the unstable Lyapunov exponents are constant across all periodic orbits, then $M$ has constant negative sectional curvature.

Viana's conjecture. If $f$ has only nonzero Lyapunov exponents at Lebesgue almost every point, then it admits some SRB measure.

Let $L$ be an irreducible dispersal matrix, let $A_i=R_i+L$ be the matrices in the differential inclusion model, and let $x_i$ be eigenvectors of $A_i$. Assume that the eigenvector…

Let $n\ge 2$ point particles of masses $m_1\ge m_2\ge\dots\ge m_n>0$ move on the vertical half line $q\ge 0$, with positions $0\le q_1\le q_2\le\dots\le q_n$, under constant gravit…

Let $M$ be a complete Riemannian manifold with finite volume, whose geodesic flow is Anosov. The periodic-exponent rigidity conjecture. If the unstable Lyapunov exponents are const…

Bochi–Katok–Rodriguez Hertz flexibility conjecture. There exists an ergodic diffeomorphism $f \in C$ such that $\xi_i$, for $i=1,\ldots,d$, are the Lyapunov exponents of $f$ with r…

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

Non-varying-stratum characterization. The stratum $\mathcal{Q}_g(d_1,\ldots,d_n)$ is non-varying if and only if $L^+$ equals the sum of the $g$ smallest numbers in $S$. This is pre…

Let $M$ be a compact manifold of any dimension, let $f:M\circlearrowleft$ be a $C^\infty$ diffeomorphism, and for $k$ define … where $\Lambda^k df$ is the action induced by $f$ on…

Let $\mathcal{H}^n$ be a weight $n$ variation of Hodge structures with thin monodromy over a hyperbolic Riemann surface $C$ with singular locus $S$. Let…

Minimizing upper-Lyapunov measure conjecture. For a generic expanding self-map $T$ on a manifold $M$, the minimizing measure of the upper Lyapunov exponent $\lambda_1$ exists, is u…

Upper Lyapunov exponent maximizing-measure conjecture. If $k\geq 2$, then for a $C^k$-generic expanding self-map $T$, the upper Lyapunov exponent has a unique maximizing measure, s…

Jenkinson–Morris conjecture. For a generic $T\in\mathcal{E}^k$, the Lyapunov minimizing measure is unique and supported on a periodic orbit of $T$.

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

Let $M$ be a compact manifold with volume measure $m$, and let $\operatorname{Diff}^{1}_{m}(M)$ be the space of volume-preserving $C^1$-diffeomorphisms of $M$. For…

Let $I$ be the phase space and let $T:I\to I$ be a $C^1$, nonsingular dynamical system admitting a unique absolutely continuous invariant measure with positive Lyapunov exponent. L…

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

Let $\hat{A}_k$ and $\hat{A}$ be continuous linear cocycles over the same hyperbolic system. Suppose that each cocycle is equipped with a uniform unstable holonomy $H^{u,k}$ and…

Let $C$ be a compact Riemann surface, and let the $n$-th Hitchin component be the connected component of … containing symmetric powers of Fuchsian representations. For a representa…

Higher-dimensional random-product conjecture. There exist a $C^1$ dense and $C^r$ open set of volume-preserving diffeomorphisms such that the random product has at least one positi…