We study random dynamical systems composed of Liverani–Saussol–Vaienti maps with varying parameters, without any mixing assumptions on the base space of random dynamics. We establish a quenched central limit theorem and identify conditions under which the associated limit variance varies continuously and differentiably with respect to perturbations of the random dynamics. Our arguments rely on recent results on statistical stability and linear response for random intermittent maps established by Dragičević et al [J. Lond. Math. Soc. 111 (2025), e70150].

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 introduce Feldman–Katok convergence for invariant measures of a topological dynamical system. This can be seen as a counterpart to the convergence with respect to the $\bar {f}$-metric for finite-state stationary processes (shift-invariant measures on a symbolic space). Feldman–Katok convergence is based on a dynamically defined Feldman–Katok pseudometric. This convergence is stronger than weak$^*$ convergence. We prove that Feldman–Katok convergence preserves ergodicity and makes the Kolmogorov–Sinai entropy lower semicontinuous, thereby preserving zero entropy. We apply our findings to non-hyperbolic (having at least one vanishing Lyapunov exponent) ergodic measures constructed using the GIKN method as axiomatized by Bonatti, Díaz and Gorodetski [Nonlinearity, 23 (2010), 687–705]. The GIKN method, originally introduced by Gorodetski, Ilyashenko, Kleptsyn and Nalsky [Functional Analysis and its Applications, 39 (2005), 21–30], has been widely adapted to produce non-hyperbolic ergodic measures for diffeomorphisms of compact manifolds. We prove that an ergodic measure satisfying the conditions provided by the axiomatized GIKN method is the Feldman–Katok limit of a sequence of periodic measures, which implies that it is either a periodic measure or a loosely Kronecker measure (a measure Kakutani equivalent to an aperiodic ergodic rotation on a compact group) and has zero entropy. This classifies all these measures up to Kakutani equivalence and confirms that geometric constructions of non-hyperbolic measures via periodic approximations based on the axiomatized GIKN method presented in Bonatti et al. [op. cit.] systematically produce zero-entropy systems.

Let $\Gamma $ be a torsion-free subgroup of $\mathrm {SL}_3(\mathbb {R})$ commensurable with $\mathrm {SL}_3(\mathbb {Z})$, and $Y=\mathrm {SO}_3(\mathbb {R})\backslash \mathrm {SL}_3(\mathbb {R})/\Gamma $ be endowed with the natural locally symmetric space structure. We prove that for any point $y\in Y$, the set of directions in which the geodesic ray starting from y is bounded in Y, is hyperplane absolute winning.

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.

We prove that, for a $C^2$ partially hyperbolic endomorphism of the 2-torus which is strongly transitive, given an ergodic u-Gibbs measure that has positive center Lyapunov exponent and has full support, then either the map is special (has only one unstable direction per point) or the measure is the unique absolutely continuous invariant measure. We can apply this result in many settings, in particular, obtaining uniqueness of u-Gibbs measures for every non-special perturbation of irreducible linear expanding maps of the torus with simple spectrum. This gives new open sets of partially hyperbolic systems displaying a unique u-Gibbs measure.

We generalize Hopf’s theorem to thermostats: the total thermostat curvature of a thermostat without conjugate points is non-positive and vanishes only if the thermostat curvature is identically zero. We further show that, if the thermostat curvature is zero, then the flow has no conjugate points and the Green bundles collapse almost everywhere. Given a thermostat without conjugate points, we prove that the Green bundles are transverse everywhere if and only if it is projectively Anosov. Finally, we provide an example showing that Hopf’s rigidity theorem on the $2$-torus cannot be extended to thermostats. It is also the first example of a projectively Anosov thermostat which is not Anosov.

We investigate the thermodynamic formalism for Viana maps—skew products obtained by coupling an expanding circle map with a slightly perturbed quadratic family on the fibers. For every Hölder potential $\varphi $ whose oscillation is below an explicit threshold, we show that an equilibrium state not only exists, but is unique and satisfies an upper level-2 large-deviation principle. All of these conclusions persist under sufficiently small perturbations of the reference map.

We introduce a correlation number for two strictly positive locally Hölder continuous independent potentials with strong entropy gaps at infinity on a topologically mixing countable state Markov shift with big images and pre-images (BIP) property. We define in this way a correlation number for pairs of cusped Hitchin representations. Furthermore, we explore the connection between the correlation number and the Manhattan curve, along with several rigidity properties of this correlation number.

We consider a DA-type surgery of the famous Lorenz attractor in dimension 4. This kind of surgery was first used by Smale [Differentiable dynamical systems. Bull. Amer. Math. Soc. (N.S.) 73(6) (1967), 747–817] and Mañé [Contributions to the stability conjecture. Topology 17(4) (1978), 383–396] to give important examples in the study of partially hyperbolic systems. Our construction gives the first example of a singular chain recurrence class which is Lyapunov stable, away from homoclinic tangencies, and exhibits robustly heterodimensional cycles. Moreover, the chain recurrence class has the following interesting property: there exists robustly a two-dimensional sectionally expanding subbundle (containing the flow direction) of the tangent bundle such that it is properly included in a subbundle of the finest dominated splitting for the tangent flow.

We construct various novel and elementary examples of dynamics with metric attractors that have intermingled basins. A main ingredient is the introduction of random walks along orbits of a given dynamical system. We develop the theory and use it in particular to provide examples of thick metric attractors with intermingled basins.

Let $f:M\to M$ be a homeomorphism over a compact Riemannian manifold, ergodic with respect to a measure $\mu $ defined on the completion of the Borel $\sigma $-algebra, and $\mathcal F$ a f-invariant one-dimensional continuous foliation of M by $C^1$-leaves. Then, if f preserves a continuous $\mathcal {F}$-arc length system, then we only have three possibilities for the conditional measures of $\mu $ along $\mathcal F$, namely: (i) they are atomic for almost every leaf; or (ii) for almost every leaf, they are equivalent to the measure $\unicode{x3bb} _x$ induced by the invariant arc-length system over $\mathcal F$; or (iii) for almost every leaf, their support is a nowhere dense, perfect subset of the leaf. Furthermore, we show that restricted to ergodic partially hyperbolic diffeomorphism with one-dimensional topological neutral center direction, we are able to eliminate the third case obtaining a dichotomy.

In this article, we study the pressure at infinity of potentials defined over countable Markov shifts. We establish an upper semi-continuity result concerning the limiting behaviour of the pressure of invariant probability measures, where the escape of mass is controlled by the pressure at infinity. As a consequence, we establish criteria for the existence of equilibrium states and maximizing measures for uniformly continuous potentials. Additionally, we study the pressure at infinity of suspension flows defined over countable Markov shifts and prove an upper semi-continuity result for the pressure map.

Given a continuous linear cocycle $\mathcal {A}$ over a homeomorphism f of a compact metric space X, we investigate its set $\mathcal {R}$ of Lyapunov-Perron regular points, that is, the collection of trajectories of f that obey the conclusions of the Multiplicative Ergodic Theorem. We obtain results roughly saying that the set $\mathcal {R}$ is of first Baire category (i.e., meager) in X, unless some rigid structure is present. In some settings, this rigid structure forces the Lyapunov exponents to be defined everywhere and to be independent of the point; that is what we call complete regularity.

We consider an infinite-dimension SIS model introduced by Delmas, Dronnier, and Zitt, with a more general incidence rate, and study its equilibria. Unsurprisingly, there exists at least one endemic equilibrium if and only if the basic reproduction number is larger than 1. When the pathogen transmission exhibits one way propagation, it is possible to observe different possible endemic equilibria. We characterize in a general setting all the equilibria, using a decomposition of the space into atoms, given by the transmission operator. We also prove that the proportion of infected individuals converges to an equilibrium, which is uniquely determined by the support of the initial condition. We extend those results to infinite-dimensional SIS models with reservoir or with immigration.

Given a weakly almost additive sequence of continuous functions with bounded variation ${\mathcal {F}}=\{\log f_n\}_{n=1}^{\infty }$ on a subshift X over finitely many symbols, we study properties of a function f on X such that $\lim _{n\to \infty }({1}/{n})\int \log f_n\,d\mu =\int f\,d\mu $ for every invariant measure $\mu $ on X. Under some conditions, we construct a function f on X explicitly, and study a relation between the property of ${\mathcal {F}}$ and some particular types of f. As applications, we study images of Gibbs measures for continuous functions under one-block factor maps. We investigate a relation between the almost additivity of the sequences associated to relative pressure functions and the fiber-wise sub-positive mixing property of a factor map. For a special type of one-block factor maps between shifts of finite type, we study necessary and sufficient conditions for the image of a one-step Markov measure to be a Gibbs measure for a continuous function.

In this paper, we study the quantitative recurrence properties in the case of $\mathbb {Z}$-extension of Axiom A flows on a Riemannian manifold. We study the asymptotic behavior of the first return time to a small neighborhood of the starting point. We establish results of almost everywhere convergence, and of convergence in distribution with respect to any probability measure absolutely continuous with respect to the infinite invariant measure. In particular, our results apply to geodesic flows on the $\mathbb {Z}$-cover of compact smooth surfaces of negative curvature.

We prove that when the Aubry set for a Lipschitz continuous potential is a subshift of finite type, then the pressure function converges exponentially fast to its asymptote as the temperature goes to 0. The speed of convergence turns out to be the unique eigenvalue for the matrix whose entries are the costs between the different irreducible pieces of the Aubry set. For a special case of Walters potential, we show that perturbations of that potential that go faster to zero than the pressure do not change the selection, neither for the subaction nor for the limit measure, a zero temperature.

We prove that a partially hyperbolic attracting set for a $C^2$ vector field, having slow recurrence to equilibria, supports an ergodic physical/SRB measure if, and only if, the trapping region admits non-uniform sectional expansion on a positive Lebesgue measure subset. Moreover, in this case, the attracting set supports at most finitely many ergodic physical/SRB measures, which are also Gibbs states along the central-unstable direction. This extends to continuous time systems a similar well-known result obtained for diffeomorphisms, encompassing the presence of equilibria accumulated by regular orbits within the attracting set. In codimension two the same result holds, assuming only the trajectories on the trapping region admit a sequence of times with asymptotical sectional expansion, on a positive volume subset. We present several examples of application, including the existence of physical measures for asymptotically sectional hyperbolic attracting sets, and obtain physical measures in an alternative unified way for many known examples: Lorenz-like and Rovella attractors, and sectional-hyperbolic attracting sets (including the multidimensional Lorenz attractor).

In this article, we consider a closed rank-one Riemannian manifold M without focal points. Let $P(t)$ be the set of free-homotopy classes containing a closed geodesic on M with length at most t, and $\# P(t)$ its cardinality. We obtain the following Margulis-type asymptotic estimates:

$$ \begin{align*}\lim_{t\to \infty}\#P(t)/\frac{e^{ht}}{ht}=1\end{align*} $$