27 problems

A flow is a continuous-time dynamical system $(\phi^t)_{t\geq 0}$, and a mixing Anosov flow is an Anosov flow whose correlations decay to zero for suitable observables. Bowen–Ruell…

Let $f:M\to M$ be a non-uniformly expanding map with an absolutely continuous invariant measure $\mu$. The map is totally expanding when every invariant measure has all Lyapunov ex…

High-temperature exponential clustering conjecture. There exist $\beta^*$, $C_\mathrm{Cov}$, $c_\mathrm{Cov}>0$, and $n\in\mathbb{N}$ such that, for every $\beta<\beta^*$ and every…

Let $\mathcal L$ be a lattice defining a periodic Lorentz process with phase space $M$, invariant measure $\mu$, free-flight time $\tau$, and flow $\Phi_t$. Let $f,g:M\to\mathbb R$…

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…

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 a flow be an Axiom A mixing flow, and let correlations be measured with respect to its invariant measure. Exponential-decay conjecture. All Axiom A mixing flows should exhibit…

Consider a generic one-parameter family of maps. Density conjecture. Maps with exponential decay of correlations are Lebesgue density points of other maps with uniform exponential…

Let $\Lambda$ be a compact forward-invariant set for a smooth map $f$. An invariant ergodic probability measure is hyperbolic if all its Lyapunov exponents are non-zero, and…

Consider smooth diffeomorphisms with hyperbolic physical measures. A stable leaf has uniform size on a full-volume subset if there is a cylinder in the ambient space on which that…

Let $r\ge1$. Consider diffeomorphisms with a dominated splitting, together with non-uniform expansion and non-uniform contraction, and with neither a uniformly expanding nor a unif…

Decay of correlations conjecture. The events satisfy $\nu_{L,m,\epsilon}^\mathrm{Lv}(\mathcal A_{x,r,L})\to1$ as first $\epsilon\to0$, then $m\to0$, and finally $L\to\infty$, and,…

S60's conjecture. Based on a numerical experiment, the decay rate of correlations for Hölder continuous functions is of polynomial order $n^{-3/2}$, and this rate is optimal.

Let $V$ be a metric space, let $\beta>0$, and let $\boldsymbol{b}:\mathcal{P}_{f}(V)\to[0,\infty)$ be a subbaditive function. Let $\Phi$ be a local interaction satisfying … For eve…

Let $F_a$ be the heterochaos baker map on $[0,1]^d$, let $\operatorname{Leb}$ denote Lebesgue measure, and let ${\rm Cor}_n(F_a;\varphi,\psi;\operatorname{Leb})$ denote the correla…

Mixing and correlation-decay conjecture. The random systems from Theorem 1.1b are mixing. Moreover, if $\Sigma_R=\{1\}$, then the rate of the annealed decay of correlations is

Let $f$ be the piecewise hyperbolic map under consideration, let $\rho$ be its SRB measure, and let $\rho_{\mathcal S}$ be the slice measure on the singular set. Write $\mathcal M$…

Exponential-mixing conjecture for Axiom A attractors. Every topologically mixing attractor of an Axiom A flow mixes exponentially with respect to all its equilibrium states.

For a Bunimovich flower, let $\rho_{v,w}(n)$ denote the correlation function for dynamically Hölder observables $v$ and $w$, with correlations taken with respect to Liouville measu…

Decay conjecture. In the infinite horizon case, the decay rate for the flow is

Let $\beta\to[0,infty]$ be the inverse temperature, let $Lambda earrow a Z^2$ be finite regions, and let $n$ denote the $ell^infty$ distance from a site $x$ to the boundary…

Correlation-decay conjecture. For $d\geq 3$, correlations restricted to trajectories with at least one collision should decay more rapidly than $C/t$; subject to this conjecture,

Consider smooth singular flows in dimensions higher than three that are sectionally-hyperbolic, meaning that they belong to the higher-dimensional class described in the source, ge…

Let a Lorenz attractor be generated by a flow of regularity $C^{1+\alpha}$. The Lorenz-attractor conjecture. $C^{1+\alpha}$ Lorenz attractors have robust exponential decay of corre…