We establish a theory for multivariate extreme value analysis of dynamical systems. Namely, we provide conditions adapted to the dynamical setting which enable the study of dependence between extreme values of the components of Rd-valued observables evaluated along the orbits of the systems. We study this cross-sectional dependence, which results from the combination of a spatial and a temporal dependence structures. We give several illustrative applications, where concrete systems and dependence sources are introduced and analysed.

We introduce dynamical blanket times, which quantify how quickly the empirical distribution along a typical orbit approximates the invariant measure. These can be viewed as a measure-theoretic analogue of the previously introduced dynamical cover time, which measures how quickly an orbit becomes dense in the space. Motivated by analogous comparability questions for random walks on graphs, we investigate how much longer it takes for a dynamical system to “blanket” than to “cover”. For finite-branch, uniformly expanding interval maps, we obtain upper bounds on the expected blanket time in terms of the spatial scale and the precision of approximation. In the special case where the invariant measure is absolutely continuous with respect to Lebesgue, this yields comparability between the expected blanket and cover times, uniformly across all sufficiently small scales. Our approach combines two main ingredients. First, we establish large deviation estimates for hitting times which are uniform over both the target location and the spatial scale. Second, using methods from multifractal analysis, we construct a finite discretisation of the invariant measure which reduces the problem to a suitably controlled discrete model.

We introduce the notion of cover time to dynamical systems. This quantifies the rate at which orbits become dense in the state space and can be viewed as a global, rather than the more standard local, notion of recurrence for a system. Using transfer operator tools for systems with holes and inducing techniques, we obtain an asymptotic formula for the expected cover time in terms of the decay rate of the measure of the ball of minimum measure. We apply this to a wide class of uniformly hyperbolic and non-uniformly hyperbolic interval maps, including the Gauss map and Manneville–Pomeau maps.

We consider local escape rates and hitting time statistics for unimodal interval maps of Misiurewicz-Thurston type. We prove that for any point z in the interval there is a local escape rate and hitting time statistics which is one of three types. While it is key that we cover all points z, the particular interest here is when z is periodic and in the postcritical orbit which yields the third part of the trichotomy. We also prove generalised asymptotic escape rates of the form first shown by Bruin, Demers and Todd.

For a probability measure preserving dynamical system (X,f,mu), the Poincar & eacute; Recurrence theorem asserts that mu -almost every orbit is recurrent with respect to its initial condition. This motivates study of the statistics of the process Xn(x)=d(fn(x),x)), and real-valued functions thereof. For a wide class of non-uniformly expanding dynamical systems, we show that the time-n counting process Rn(x) associated to the number recurrences below a certain radii sequence rn(tau) follows an averaged Poisson distribution G(tau). Furthermore, we obtain quantitative results on almost sure rates for the recurrence statistics of the process Xn.

We prove functional limit theorems for stochastic processes which have clusters of large values which, when summed and suitably normalised in time and space, get collapsed in a jump of the limiting process observed at the same time point. In order to keep track of the clustering information, which gets lost in the usual Skorohod topologies in the space of c\`adl`ag functions, we introduce a new space specially designed for that purpose which generalises the already more general spaces introduced by Whitt. Our main applications are to hyperbolic and non-uniformly expanding dynamical systems with heavy tailed observable functions maximised at dynamically linked maximal sets (such as periodic points). We also study limits of extremal processes and record times point processes for observables not necessarily heavy tailed. The applications studied include hyperbolic systems such as Anosov diffeomorphisms, but also non-uniformly expanding maps such as maps with critical points of Benedicks-Carleson type or indifferent fixed points such as Manneville-Pomeau maps. The main tool is a limit theorem for point processes with decorations derived from a bi-infinite sequence which we call the piling process, designed to keep the clustering information. This piling process is based on the tail process defined in the regularly varying context rather than the general stationary stochastic processes we consider here.

For a class of potentials $\psi$ satisfying a condition depending on the roof function of a suspension (semi)flow, we show an EKP inequality, which can be interpreted as a H\"older continuity property in the weak${^*}$ norm of measures, with respect to the pressure of those measures, where the H\"older exponent depends on the $L^q$-space that $\psi$ belongs to. This also captures a new type of phase transition for intermittent (semi)flows (and maps).

We consider interval maps with countably many full branches and observables with polynomial tails. We show that the Birkhoff spectrum is real analytic and that its convergence to the Hausdorff dimension of the repeller is governed by the polynomial tail exponent. This result extends previous work by Arima on more regular observables and demonstrates how the tail behaviour influences the structure of the Birkhoff spectrum. Our proof relies on techniques from thermodynamic formalism and tail estimates for the observable and our applications are to natural observations on Gauss maps, Lüroth transformations as well as to a the first return time for a class of induced Manneville-Pomeau maps.

We consider heavy-tailed observables maximised on a dynamically defined Cantor set and prove convergence of the associated point processes as well as functional limit theorems. The Cantor structure, and its connection to the dynamics, causes clustering of large observations: this is captured in the `decorations' on our point processes and functional limits, an application of the theory developed in a paper by the latter three authors.

We present a general framework for weak convergence to decorated L\'evy processes in enriched spaces of c\`adl\`ag functions for vector-valued processes arising in deterministic systems. Applications include uniformly expanding maps and unbounded observables as well as nonuniformly expanding/hyperbolic maps with bounded observables. The latter includes intermittent maps and dispersing billiards with flat cusps. In many of these examples, convergence fails in all of the Skorohod topologies. Moreover, the enriched space picks up details of excursions that are not recorded by Skorohod or Whitt topologies.

We give a set of equivalent conditions for a potential on a Countable Markov Shift to have strong positive recurrence, which is also equivalent to having exponential decay of correlations. A key ingredient of our proofs is quantifying how the shift behaves at its boundary.

AbstractGiven a dynamical system, we prove that the shortest distance between two n-orbits scales like n to a power even when the system has slow mixing properties, thus building and improving on results of Barros, Liao and the first author [On the shortest distance between orbits and the longest common substring problem. Adv. Math.344 (2019), 311–339]. We also extend these results to flows. Finally, we give an example for which the shortest distance between two orbits has no scaling limit.

We study continuity, and lack thereof, of thermodynamical properties for onedimensional dynamical systems. Under quite general hypotheses, the free energy is shown to be almost upper-semicontinuous: some normalised component of a limit measure will have free energy at least that of the limit of the free energies. From this, we deduce results concerning existence and continuity of equilibrium states (including statistical stability). Metric entropy, not semicontinuous as a general multimodal map varies, is shown to be upper semicontinuous under an appropriate hypothesis on critical orbits. Equilibrium states vary continuously, under mild hypotheses, as one varies the parameter and the map. We give a general method for constructing induced maps which automatically give strong exponential tail estimates. This also allows us to recover, and further generalise, recent results concerning statistical properties (decay of correlations, etc.). Counterexamples to statistical stability are given which also show sharpness of the main results.

We study continuity, and lack thereof, of thermodynamical properties for one-dimensional dynamical systems. Under quite general hypotheses, the free energy is shown to be almost upper-semicontinuous: some normalised component of a limit measure will have free energy at least that of the limit of the free energies. From this, we deduce results concerning existence and continuity of equilibrium states (statistical stability). Counterexamples to statistical stability in the absence of strong hypotheses are provided.

Regularity properties of the pressure are related to phase transitions. In this article we study thermodynamic formalism for systems defined in non-compact phase spaces, our main focus being countable Markov shifts. We produce metric compactifications of the space which allow us to prove that the pressure is differentiable on a residual set and outside an Aronszajn null set in the space of uniformly continuous functions. We establish a criterion, the so-called sectorially arranged property, which implies that the pressure in the original system and in the compactification coincide. Examples showing that the compactifications can have rich boundaries, for example a Cantor set, are provided.

Given a one-dimensional dynamical system we study its cover time, which quantifies the rate at which orbits become dense in the state space. Using transfer operator tools for dynamical systems with holes and inducing techniques, for a wide class of uniformly hyperbolic and non-uniformly hyperbolic systems we obtain an asymptotic formula for the expected cover time in terms of the decay rate of the measure of the ball of minimum measure. Applications include the Gauss map and Manneville-Pomeau maps.

In this paper we study ergodic theory of countable Markov shifts. These are dynamical systems defined over non-compact spaces. Our main result relates the escape of mass, the measure theoretic entropy, and the entropy at infinity of the system. This relation has several consequences. For example we obtain that the entropy map is upper semi-continuous and that the ergodic measures form an entropy dense subset. Our results also provide new proofs of results describing the existence and stability of the measure of maximal entropy. We relate the entropy at infinity with the Hausdorff dimension of the set of recurrent points that escape on average. Of independent interest, we prove a version of Katok's entropy formula in this non-compact setting.

We obtain limit theorems (Stable Laws and Central Limit Theorems, both standard and non-standard) and thermodynamic properties for a class of non-uniformly hyperbolic flows: almost Anosov flows, constructed here. The link between the pressure function and limit theorems is studied in an abstract functional analytic framework, which may be applicable to other classes of non-uniformly hyperbolic flows.

We consider multimodal maps with holes and study the evolution of the open systems with respect to equilibrium states for both geometric and Hölder potentials. For small holes, we show that a large class of initial distributions share the same escape rate and converge to a unique absolutely continuous conditionally invariant measure; we also prove a variational principle connecting the escape rate to the pressure on the survivor set, with no conditions on the placement of the hole. Finally, introducing a weak condition on the centre of the hole, we prove scaling limits for the escape rate for holes centred at both periodic and non-periodic points, as the diameter of the hole goes to zero.

We prove that the entropy map for countable Markov shifts of finite entropy is upper semi-continuous at ergodic measures. Note that the phase space is non-compact. Applications to systems that can be coded by these shifts, such as positive entropy diffeomorphisms on compact manifolds, are given. We also discuss the related problem of existence of measures of maximal entropy.