21 problems

Physical-measure conjecture. The attractors constructed in Section admit a unique physical/SRB measure with a full volume ergodic basin.

Let $\mathcal{U}$ be the open family of maps considered in the paper, and let $f_n$ be $C^\infty$ dynamics in $\mathcal{U}$ with physical measures $\mu_n$. Suppose that $f_n$ conve…

Let $\mathcal{D}$ be the set of circle local homeomorphisms $f:\mathbb{S}^1\to\mathbb{S}^1$ that are topologically conjugate to the doubling map. For $f\in\mathcal{D}$, let $p$ be…

Physical/SRB without slow recurrence conjecture. In Theorem and Theorem the slow recurrence conditions are superfluous.

Let $f$ be a smooth diffeomorphism with a hyperbolic $cu$-Gibbs state. Its geometric basin is the set of points whose empirical measures accumulate on that state, while its ergodic…

Let $f$ be a diffeomorphism and let the attracting set under consideration have a dominated splitting. Weak non-uniform expansion means that … on a positive-volume subset of points…

Let $\varphi_{\alpha,b}$ be one of the maps in the setting of Theorem; an SRB/physical measure is a measure with the corresponding SRB or physical property. Uniqueness conjecture.…

Let a dynamical system have a positive-volume subset of its ambient space. A point is $p\omega$-regular if it has the regularity property denoted by $p\omega$ in the source; forwar…

Let $G$ be a $C^2$ vector field with a partially hyperbolic attracting set $\Lambda=\Lambda_G(U)$, and suppose that the non-uniform sectional expansion condition holds on a positiv…

Let $G$ be a $C^2$ vector field with a partially hyperbolic attracting set $\Lambda=\Lambda_G(U)$, and let $\Omega\subset U$ be a positive-volume subset on which the weak non-unifo…

Consider generic $C^1$ vector fields having a sectional-hyperbolic attracting set. Let the bounds referred to in Theorem and Corollary be the corresponding bounds for the number of…

Let a sectional-hyperbolic attracting set be an attracting set with stable and center-unstable dimensions satisfying $d_s\geq 1$ and $d_{cu}\geq 2$, and replace Lorenz-like singula…

Consider the family of $3$-vector fields $G$ of class $C^2$ exhibiting a singular-hyperbolic attracting set containing some Lorenz-like singularity. Write $s$ for the number of erg…

Let $\mathcal{V}$ be the open family of $C^2$ vector fields admitting a singular-hyperbolic attracting set $\Lambda$. For each $G\in\mathcal{V}$, let $s(G)\in\mathbb{Z}_0^+$ be the…

Let $M$ be the compact manifold underlying the space of diffeomorphisms, and let $\operatorname{Diff}^1(M)$ denote the space of $C^1$ diffeomorphisms of $M$. A saddle-type hyperbol…

Basin equality conjecture. For any sectional-hyperbolic attracting set $\Lambda$, the topological basin of attraction coincides with the family of stable manifolds through the poin…

Let $\phi$ be a ${{\mathcal C}^\infty}$ Cherry flow on the torus $\mathbb{T}^2$ without closed orbits and with hyperbolic saddle and repelling singularities. Let…

Let $g_Y:I_\varepsilon\times\mathbb{T}^k\circlearrowleft$ be a non-skew-product quotient map, and let $P_0$ be the sink. Assume that the image of $P_2(Y)$ under the return map…

Let $X$ be the vector field under consideration and let $Y$ be a vector field $C^2$ close to $X$. Let $g_Y$ be the quotient map, let $\mu_Y$ denote a candidate physical measure, an…

Let $X$ be the vector field from Theorem 3.2, let $\mathcal{U}$ be its neighborhood in $\mathfrak{X}^2(M)$, and for $Y\in\mathcal{U}$ let $g_Y$ be the quotient map. Let $p(Y)$ be t…

Let $X$ be the vector field from Theorem 3.2 and let $\mathcal{U}$ be the neighborhood of $X$ in the space of $C^2$ vector fields. For $Y\in\mathcal{U}$, let $g_Y$ be the quotient…