We describe a new construction of equilibrium states for a class of partially hyperbolic systems. This generalizes our construction for Gibbs measures in the uniformly hyperbolic setting. This more general setting introduces new issues that we need to address carefully, in particular requiring additional assumptions on the transformation. We treat two cases: either the centre-stable manifold satisfies a bounded expansion condition; or the centre-unstable manifold satisfies a subexponential contraction condition which appears new in the context of equilibrium state constructions. The problem of constructing equilibrium states was previously raised by Pesin and Sinai and by Dolgopyat for the particular case of u-Gibbs measures, and by Climenhaga, Pesin and Zelerowicz for other equilibrium states.

Given a Fuchsian group $\Gamma $, we introduce escape rate spectra associated with $\Gamma $ to investigate the transient behavior of the geodesic flow on the hyperbolic surface $\mathbb {D}/\Gamma $. Our definition is motivated by Bishop’s linear escape set. In the special case when $\mathbb {D}/\Gamma $ is a hyperbolic $\mathbb {Z}$-covering of a surface uniformized by a generalized Schottky group, we completely determine the escape rate spectra by the convex conjugate of a generalized Poincaré exponent.

We study the probability that an AR(1) Markov chain $X_{n+1}=aX_n+\xi _{n+1}$, where $a\in (0,1)$ is a constant, stays non-negative for a long time. We find the exact asymptotics of this probability and the weak limit of $X_n$ conditioned to stay non-negative, assuming that the independent and identically distributed innovations $\xi _n$ take only two values $\pm 1$ and $a \le \tfrac 23$. This limiting distribution is quasi-stationary. It has no atoms and is singular with respect to the Lebesgue measure when $\tfrac 12< a \le \tfrac 23$, except for the case when $a=\tfrac 23$ and $\mathbb P(\xi _n=1)=\tfrac 12$, where this distribution is uniform on the interval $[0,3]$. This is similar to the properties of Bernoulli convolutions. For $0 < a \le \tfrac 12$, the situation is much simpler and the limiting distribution is a $\delta $-measure. To prove these results, we uncover a close connection between $X_n$ killed at exiting $[0, \infty )$ and the classical dynamical system defined by the piecewise linear mapping $x \mapsto x/a + 1/2\ \pmod 1$. Namely, the trajectory of this system started at $X_n$ deterministically recovers the values of the killed chain in reversed time. We use this fact to construct a suitable Banach space, where the transition operator of the killed chain has the compactness properties that allow us to apply a conventional argument of the Perron–Frobenius type.

We prove that a formal curve $\Gamma $ that is invariant by a $C^{\infty }$ vector field $\xi $ of $\mathbb R^{m}$ has a geometrical realization, as soon as the Taylor expansion of $\xi $ is not identically zero along $\Gamma $. This means that there is a trajectory $\gamma \subset \mathbb R^{m}$ of $\xi $ which is asymptotic to $\Gamma $. This result solves a natural question proposed by Bonckaert [Smooth invariant curves of singularities of vector fields in R3. Ann. Inst. Henri Poincaré 3(2) (1986), 111–183] nearly forty years ago. We also construct an invariant $C^0$ manifold S in some open horn around $\Gamma $ which is composed entirely of trajectories asymptotic to $\Gamma $ and contains the germ of any such trajectory. If $\xi $ is analytic, we prove that there exists a trajectory $\gamma $ asymptotic to $\Gamma $ which is, moreover, non-oscillating with respect to subanalytic sets.