
## 5. Other Directions

1. #### Problem 5.1.

[Kurt Vinhage] Let $f_1,f_2$ be commuting hyperbolic diffeomorphisms of a compact manifold $M$. One may construct coarse Lyapunov foliations $\mathcal W$. Given a potential $\varphi$, can one construct leaf-wise conditional measures $\mu^{\mathcal W}_\varphi$ which satisfy Margulis-like cocycle properties?
• #### Problem 5.2.

[Dan Thompson] For metric Anosov flows (Smale flows), under what conditions does orbit equivalence imply Hölder orbit equivalence? Can one find a Smale flow which is not Hölder covered by the suspension of a symbolic dynamical system.
• #### Problem 5.3.

[Kurt Vinhage] Let $f : \mathbb{T}^2 \to \mathbb{T}^2$ be a nonuniformly hyperbolic diffeomorphism. Let $\mu$ be the unstable SRB measure, $\nu$ be the measure of maximal entropy and $m$ be the measure of maximal dimension (make assumptions on $f$ so that these exist and are unique). If $\nu = \mu$ or $m$, is $f$ smoothly conjugate to a linear automorphism?
• #### Problem 5.4.

[Jayadev Athreya] Can one prove exponential decay of correlations for equilibrium states in settings where it it is known for the measure of maximal entropy? (eg, Weil-Petersson geodesic flows, noncompact spaces, non-positive curvature, etc)
• #### Problem 5.5.

[Jerome Buzzi] Can one find a dynamical interpretation of the pressure gap? For instance, can one relate the pressure gap to an essential spectral radius for a dynamical operator? Or some dynamically defined $\zeta$-function?
• #### Problem 5.6.

[Kiho Park] Describe the set of matrix cocycles which have multiple eqilibrium states. For simplicity, consider a subshift of finite type, and a locally constant, fiber bunched cocycle.
• #### Problem 5.7.

[Vaughn Climenhaga] build a moduli space for $C^{1,\theta}$-perturbations of geodesic flows using potential functions (following McMullen’s approach of expanding maps of the circle). In this case, can one identify the space of geodesic flows by a condition on their corresponding potential function?
• #### Problem 5.8.

[Mark Demers] Consider a finite-horizon billiards, and let $\varphi$ be the geometric potential. Is there a jump in pressure $P(q\varphi)$ and $q \to 0^-$? Can one find perioidic orbits with arbitrarily large (but finite) Lyapunov exponent?
• #### Problem 5.9.

[Kiho Park] Develop the theory for weighting orbits in a subadditive (rather than additive) way. {m ̊Ask Kiho for references}

Cite this as: AimPL: Equilibrium states for dynamical systems arising from geometry, available at http://aimpl.org/equibdynsysgeom.