In this survey we offer an overview of the so-called local entropy theory, which has been in development since the early 1990s. While doing so, we emphasize the connections between the topological dynamics and the ergodic theory points of view.

A space X is said to be almost totally disconnected if the set of its degenerate components is dense in X. We prove that an almost totally disconnected compact metric space admits a minimal map if and only if either it is a finite set or it has no isolated point. As a consequence we obtain a characterization of minimal sets on dendrites and local dendrites.

Let f:M→ℝ be a Morse–Bott function on a compact smooth finite-dimensional manifold M. The polynomial Morse inequalities and an explicit perturbation of f defined using Morse functions fj on the critical submanifolds Cj of f show immediately that MBt(f)=Pt(M)+(1+t)R(t), where MBt(f) is the Morse–Bott polynomial of f and Pt(M) is the Poincaré polynomial of M. We prove that R(t) is a polynomial with non-negative integer coefficients by showing that the number of gradient flow lines of the perturbation of f between two critical points p,q∈Cj of relative index one coincides with the number of gradient flow lines between p and q of the Morse function fj. This leads to a relationship between the kernels of the Morse–Smale–Witten boundary operators associated to the Morse functions fj and the perturbation of f. This method works when M and all the critical submanifolds are oriented or when ℤ2 coefficients are used.

We extend the weakly mixing PET (polynomial ergodic theorem) obtained in Bergelson [Weakly mixing PET. Ergod. Th. & Dynam. Sys.7 (1987), 337–349] to much wider families of functions. Besides throwing new light on the question of ‘how much higher-degree mixing is hidden in weak mixing’, the obtained results also show the way to possible new extensions of the polynomial Szemerédi theorem obtained in Bergelson and Leibman [Polynomial extensions of van der Waerden’s and Szemerédi’s theorems. J. Amer. Math. Soc.9 (1996), 725–753].

Let β∈(1,2). Each x∈[0,1/(β−1)] can be represented in the form where εk∈{0,1} for all k (a β-expansion of x). If , then, as is well known, there always exist x∈(0,1/(β−1)) which have a unique β-expansion. We study (purely) periodic unique β-expansions and show that for each n≥2 there exists such that there are no unique periodic β-expansions of smallest period n for β≤βn and at least one such expansion for β>βn. Furthermore, we prove that βk<βm if and only if k is less than m in the sense of the Sharkovskiĭ ordering. We give two proofs of this result, one of which is independent, and the other one links it to the dynamics of a family of trapezoidal maps.

We show that the ordered Matsumoto K0-group of a substitutive system is derivable from the ordered Čech cohomology of a simple ‘augmentation’ of the tiling space associated with the substitution. Besides supplying a topological interpretation of the Matsumoto K0-group, this leads to finer invariants of the flow equivalence class of a substitutive system.

Interval exchange maps are related to geodesic flows on translation surfaces; they correspond to the first return maps of the vertical flow on a transverse segment. The Rauzy–Veech induction on the space of interval exchange maps provides a powerful tool to analyze the Teichmüller geodesic flow on the moduli space of Abelian differentials. Several major results have been proved using this renormalization. Danthony and Nogueira introduced in 1988 a natural generalization of interval exchange transformations, namely linear involutions. These maps are related to general measured foliations on surfaces (whether orientable or not). In this paper we are interested by such maps related to geodesic flow on (orientable) flat surfaces with ℤ/2ℤ linear holonomy. We relate geometry and dynamics of such maps to the combinatorics of generalized permutations. We study an analogue of the Rauzy–Veech induction and give an efficient combinatorial characterization of its attractors. We establish a natural bijection between the extended Rauzy classes of generalized permutations and connected components of the strata of meromorphic quadratic differentials with at most simple poles, which allows us, in particular, to classify the connected components of all exceptional strata.

Downarowicz [Entropy structure. J. Anal.96 (2005), 57–116] stated a variational principle for the tail entropy for invertible continuous dynamical systems of a compact metric space. We give here an elementary proof of this variational principle. Furthermore, we extend the result to the non-invertible case.

A homeomorphism f of a manifold M is called H1-transitive if there is a transitive lift of an iterate of f to the universal Abelian cover . Roughly speaking, this means that f has orbits which repeatedly and densely explore all elements of H1(M). For a rel pseudo-Anosov map ϕ of a compact surface M we show that the following are equivalent: (a) ϕ is H1-transitive, (b) the action of ϕ on H1(M) has spectral radius one and (c) the lifts of the invariant foliations of ϕ to have dense leaves. The proof relies on a characterization of transitivity for twisted ℤd-extensions of a transitive subshift of finite type.

We provide effective versions of theorems of Furstenberg and Rudolph–Johnson regarding closed subsets and probability measures of ℝ/ℤ invariant under the action of a non-lacunary multiplicative semigroup of integers. In particular, we give an explicit rate at which the sequence {anbkx}n,k becomes dense for a,b fixed multiplicatively independent integers and x∈ℝ/ℤ Diophantine generic.

We study aperiodic substitution dynamical systems arising from non-primitive substitutions. We prove that the Vershik homeomorphism φ of a stationary ordered Bratteli diagram is topologically conjugate to an aperiodic substitution system if and only if no restriction of φ to a minimal component is conjugate to an odometer. We also show that every aperiodic substitution system generated by a substitution with nesting property is conjugate to the Vershik map of a stationary ordered Bratteli diagram. It is proved that every aperiodic substitution system is recognizable. The classes of m-primitive substitutions and derivative substitutions associated with them are studied. We discuss also the notion of expansiveness for Cantor dynamical systems of finite rank.

Gibbs measures μ on cookie-cutter sets are the archetype of multifractal measures on Cantor sets. We compute the singularity spectrum of the inverse measure of μ. Such a measure is discrete (it is constituted only by Dirac masses), it satisfies a multifractal formalism and its Lq-spectrum possesses one point of non-differentiability. The results rely on heterogeneous ubiquity theorems.

Assume that x∈[0,1) admits its continued fraction expansion x=[a1(x),a2(x),…]. The Khintchine exponent γ(x) of x is defined by when the limit exists. The Khintchine spectrum dim Eξ is studied in detail, where Eξ:={x∈[0,1):γ(x)=ξ}(ξ≥0) and dim denotes the Hausdorff dimension. In particular, we prove the remarkable fact that the Khintchine spectrum dim Eξ, as a function of , is neither concave nor convex. This is a new phenomenon from the usual point of view of multifractal analysis. Fast Khintchine exponents defined by are also studied, where φ(n) tends to infinity faster than n does. Under some regular conditions on φ, it is proved that the fast Khintchine spectrum dim ({x∈[0,1]:γφ(x)=ξ}) is a constant function. Our method also works for other spectra such as the Lyapunov spectrum and the fast Lyapunov spectrum.

Let M be a smooth compact Riemannian manifold without boundary, and ϕ:M×ℝ→M a transitive Anosov flow. We prove that if the time-one map of ϕ is C1-approximated by Axiom-A diffeomorphisms with more than one attractor, then ϕ is topologically equivalent to the suspension of an Anosov diffeomorphism.

We prove the existence of the limiting distribution for the sequence of denominators generated by continued fraction expansions with even partial quotients, which were introduced by Schweiger [Continued fractions with odd and even partial quotients. Arbeitsberichte Math. Institut Universtät Salzburg4 (1982), 59–70; On the approximation by continues fractions with odd and even partial quotients. Arbeitsberichte Math. Institut Universtät Salzburg1–2 (1984), 105–114] and studied also by Kraaikamp and Lopes [The theta group and the continued fraction expansion with even partial quotients. Geom. Dedicata59(3) (1996), 293–333]. Our main result is proven following the strategy used by Sinai and Ulcigrai [Renewal-type limit theorem for the Gauss map and continued fractions. Ergod. Th. & Dynam. Sys.28 (2008), 643–655] in their proof of a similar renewal-type theorem for Euclidean continued fraction expansions and the Gauss map. The main steps in our proof are the construction of a natural extension of a Gauss-like map and the proof of mixing of a related special flow.

We study the dynamics of piecewise affine surface homeomorphisms from the point of view of their entropy. Under the assumption of positive topological entropy, we establish the existence of finitely many ergodic and invariant probability measures maximizing entropy and prove a multiplicative lower bound for the number of periodic points. This is intended as a step towards the understanding of surface diffeomorphisms. We proceed by building a jump transformation, using not first returns but carefully selected ‘good’ returns to dispense with Markov partitions. We control these good returns through some entropy and ergodic arguments.

We analyze in detail the notions of induction and restriction for cellular automata. As a by-product we extend a few classical and recent theorems on cellular automata to uncountable groups.

For a map of the unit interval with an indifferent fixed point, we prove an upper bound for the variance of all observables of n variables, K:[0,1]n→ℝ, which are separately Lipschitz. The proof is based on coupling and decay of correlation properties of the map. We also present applications of this inequality to the almost-sure central limit theorem, the kernel density estimation, the empirical measure and the periodogram.

A measure-preserving isomorphism α of a probability space (Ω,ℱ,μ) generates an algebraic automorphism of the algebra of bounded linear operators in 𝕃p(Ω,ℱ,μ). An ergodic theorem for is proved.

In 2004, Féjoz [Démonstration du ‘théoréme d’Arnold’ sur la stabilité du système planétaire (d’après M. Herman). Ergod. Th. & Dynam. Sys.24(5) (2004), 1521–1582], completing investigations of Herman’s [Démonstration d’un théoréme de V.I. Arnold. Séminaire de Systémes Dynamiques et manuscripts, 1998], gave a complete proof of ‘Arnold’s Theorem’ [V. I. Arnol’d. Small denominators and problems of stability of motion in classical and celestial mechanics. Uspekhi Mat. Nauk. 18(6(114)) (1963), 91–192] on the planetary many-body problem, establishing, in particular, the existence of a positive measure set of smooth (C∞) Lagrangian invariant tori for the planetary many-body problem. Here, using Rüßmann’s 2001 KAM theory [H. Rüßmann. Invariant tori in non-degenerate nearly integrable Hamiltonian systems. R. & C. Dynamics2(6) (2001), 119–203], we prove the above result in the real-analytic class.