
## 3. Specific Examples

1. #### Problem 3.1.

[Keith Burns] Let $\varphi_1,\varphi_2 : X \to \R$ be two potentials. What interesting analysis can be said about the 2-parameter family of equilibrium states $\mu_{q_1\varphi_1 + q_2\varphi_2}$? Can we find useful applications for these results?
• #### Problem 3.2.

[Todd Fisher] Generalize the results of Bufetov and Gurevich in [MR2857792] to the space of quadratic differentials.
• #### Problem 3.3.

[Todd Fisher and Keith Burns] What can be said about interesting phase transitions for potentials $\{q\varphi\}_{q\in\R}$? In particular, let $M$ be a compact surface of nonpositive curvature, and $\varphi$ be the geometric potential, so that $q = 1$ is a phase transition.
• #### Problem 3.4.

[Yuri Lima] Consider a dynamical billiard without cusps. Do (non-periodic) invariant probability measures for the billiard map satisfy the integrability condition for Oseledet’s theorem?
• #### Problem 3.5.

[Vaughn Climenhaga] Are there examples of non-uniformly hyperbolic magnetic flows for which we can apply the tools of thermodynamical formalism?
• #### Problem 3.6.

[Kurt Vinhage] Study the measure of maximal entropy for the Handel-Thurston Anosov flow. Does it coincide with Lebesgue measure? If not, what is its dimension?
• #### Problem 3.7.

[Jermoe Buzzi] Can one identify a geometric example of a system for which the set of phase transitions has positive Lebesgue measure?

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