Topological pressure for conservative C¹-diffeomorphisms with no dominated splitting

Abstract

We prove three formulas for computing the topological pressure of $C^1$-generic conservative diffeomorphism with no dominated splitting and show the continuity of topological pressure with respect to these diffeomorphisms. We prove for these generic diffeomorphisms that there are no equilibrium states with positive measure theoretic entropy. In particular, for hyperbolic potentials, there are no equilibrium states. For $C^1$ generic conservative diffeomorphism on compact surfaces with no dominated splitting and ${\varphi }_m\left(x\right):=-\frac{1}{m}\log \Vert D_x{f}^m\Vert ,m\in \mathbb{N}$, we show that there exist equilibrium states with zero entropy and there exists a transition point $t_0$ for the one parameter family $\{t{\varphi }_m{\}}_{t\ge 0}$, such that there is no equilibrium states for $t\in [0,t_0)$ and there is an equilibrium state for $t\in [t_0,+\mathrm{\infty })$.

Keywords:

• Topological pressure
• measure theoretic entropy
• dominated splitting
• equilibrium states
• phase transition

Acknowledgments

The author would like to thank Todd Fisher for introducing this problem to him and for useful discussions, Sylvain Crovisier for answering some questions about the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 One can find a diffeomorphism such that for each $1\le i<d_0$ there exists a periodic point p with period m such that $D_p{f}^m$ has $d_0$ simple eigenvalues ${\lambda }_1,\dots ,{\lambda }_{d_0}$ such that $|{\lambda }_k|<|{\lambda }_{k+1}|$ for each $k\ne i$ and such that ${\lambda }_i,{\lambda }_{i+1}$ are non-real conjugated complex numbers. Then if f has a dominated splitting on the entire manifold, by continuity of the splitting, we have the dimensions of the finest splitting will be constant which contradicts our construction.

2 For an area-preserving diffeomorphism, if it has a dominated splitting on a compact invariant set, then the splitting is hyperbolic. So ${\mathcal{E}}_{\omega }\left(M\right)$ is the interior of the set of all non-Anosov diffeomorphisms on M.