We show some level-2 large deviation principles for rational maps satisfying a strong form of non-uniform hyperbolicity, called ‘Topological Collet–Eckmann’. More precisely, we prove a large deviation principle for the distribution of iterated preimages, periodic points, and Birkhoff averages. For this purpose we show that each Hölder continuous potential admits a unique equilibrium state, and that the pressure function can be characterized in terms of iterated preimages, periodic points, and Birkhoff averages. Then we use a variant of a general result of Kifer.

We consider an ergodic process on finitely many states, with positive entropy. Our main result asserts that the distribution function of the normalized waiting time for the first visit to a small (i.e., over a long block) cylinder set B is, for majority of such cylinders and up to epsilon, dominated by the exponential distribution function 1−e−t. That is, the occurrences of B along the time axis can appear either with gap sizes of nearly the exponential distribution (as in an independent and identically distributed process), or they attract each other and create ‘series’. We recall that in [T. Downarowicz, Y. Lacroix and D. Leandri. Spontaneous clustering in theoretical and some empirical stochastic processes. ESAIM Probab. Stat. to appear] it is proved that in a typical (in the sense of category) ergodic process (of any entropy), all cylinders B of selected lengths (such lengths have upper density 1 in ℕ) reveal strong attracting. Combining this with the result of this paper, we obtain, globally in ergodic processes of positive entropy and for long cylinder sets, the prevalence of attracting and deficiency of repelling. This phenomenon resembles what in real life is known as the law of series; the common-sense observation that a rare event, having occurred, has a mysterious tendency to untimely repetitions.

We show how multifractal properties of a measure supported by a fractal F⊆[0,1] may be expressed in terms of complementary intervals of F and thus in terms of spectral triples and the Dixmier trace of certain operators. For self-similar measures this leads to a non-commutative integral over F equivalent to integration with respect to an auxiliary multifractal measure.

Mean dimension is an invariant which makes it possible to distinguish between topological dynamical systems with infinite entropy. Extending in part the work of Lindenstrauss we show that if (X,ℤk) has a free zero-dimensional factor then it can be embedded in the ℤk-shift on ([0,1]d)ℤk, where d=[C(k) mdim(X,ℤk)]+1 for some universal constant C(k), and a topological version of the Rokhlin lemma holds. Furthermore, under the same assumptions, if mdim(X,ℤk)=0, then (X,ℤk) has the small boundary property. One of the applications of this theory is related to Downarowicz’s entropy structure, a master invariant for entropy theory, which captures the emergence of entropy on different scales. Indeed, we generalize this invariant and prove the Boyle–Downarowicz symbolic extension entropy theorem in the setting of ℤk-actions. This theorem describes what entropies are achievable in symbolic extensions.

Given a Polish group G of isometries of a locally compact separable metric space, we prove that each measure-preserving Boolean action by G has a spatial model or, in other words, has a point realization. This result extends both a classical theorem of Mackey and a recent theorem of Glasner and Weiss, and it covers interesting new examples. In order to prove our result, we give a characterization of Polish groups of isometries of locally compact separable metric spaces which may be of independent interest. The solution to Hilbert’s fifth problem plays an important role in establishing this characterization.

In this article we prove the existence and uniqueness of equilibrium states for the potential and the class of non-uniformly hyperbolic horseshoes which was introduced in Rios [Unfolding homoclinic tangencies inside horseshoes: hyperbolicity, fractal dimensions and persistent tangencies. Nonlinearity14 (2001), 431–462]. We show that the pressure t↦𝒫(t) for −tlog Ju is real-analytic on . We give the exact equations of the two asymptotes to the graph of 𝒫(t) at ±∞ and we prove that these non-uniformly hyperbolic horseshoes do not have measures which minimize the unstable Lyapunov exponent.

This paper presents an example of Riemann surface lamination with at least two ergodic invariant measures. The generic leaves for those measures are of different growth and have different numbers of ends.

We present a class of subshifts ZN,N=1,2,…, whose associated C*-algebras 𝒪ZN are simple, purely infinite and not stably isomorphic to any Cuntz–Krieger algebra nor to the Cuntz algebra 𝒪∞. The class of the subshifts is the first example whose associated C*-algebras are not stably isomorphic to any Cuntz–Krieger algebra nor to the Cuntz algebra 𝒪∞. The subshifts ZN are coded systems whose languages are context free. We compute the topological entropy for the subshifts and show that a KMS-state (a state satisfying the Kubo–Martin–Schwinger condition) for gauge action on the associated C*-algebra 𝒪ZN exists if and only if the logarithm of the inverse temperature is the topological entropy for the subshift ZN, and the corresponding KMS-state is unique.

In this paper, we study perturbations of multidimensional shifts of finite type. Specifically, for any ℤd shift of finite type X with d>1 and any finite pattern w in the language of X, we denote by Xw the set of elements of X not containing w. For strongly irreducible X and patterns w with shape a d-dimensional cube, we obtain upper and lower bounds on htop (X)−htop (Xw) dependent on the size of w. This extends a result of Lind for d=1 . We also apply our methods to an undecidability question in ℤd symbolic dynamics.

We give a new definition of topological pressure for arbitrary (non-compact, non-invariant) Borel subsets of metric spaces. This new quantity is defined via a suitable variational principle, leading to an alternative definition of an equilibrium state. We study the properties of this new quantity and compare it with existing notions of topological pressure. We are particularly interested in the situation when the ambient metric space is assumed to be compact. We motivate our definition by applying it to some interesting examples, including the level sets of the pointwise Lyapunov exponent for the Manneville–Pomeau family of maps.

We show that every locally finite random graph embedded in the plane with an isometry-invariant distribution can be five-colored in an invariant and deterministic way, under some non-triviality assumption and a mild assumption on the tail of edge lengths. The assumptions hold for any Voronoi map on a point process that has no non-trivial symmetries almost surely, hence we improve and generalize previous results on six-coloring the Voronoi map on a Poisson point process (see Angel, Benjamini, Gurel-Gurevich, Mayerovitch and Peled [Stationary map coloring. Preprint, 2008]).

We provide examples of two-dimensional diffeomorphisms which cannot be topologically conjugate to any diffeomorphism of a higher smoothness.

For any regular Markov operator on the space of finite Borel measures on a Polish space we give a Yosida-type decomposition of the state space, which yields a parametrization of the ergodic probability measures associated with this operator in terms of particular subsets of the state space. We use this parametrization to prove an integral decomposition of every invariant probability measure in terms of the ergodic probability measures and give an ergodic decomposition of the state space. This extends results by Yosida [Functional Analysis. Springer, Berlin, 1980, Ch. XIII.4], Hernández-Lerma and Lasserre [Ergodic theorems and ergodic decomposition for Markov chains. Acta Appl. Math.54 (1998), 99–119] and Zaharopol [An ergodic decomposition defined by transition probabilities. Acta Appl. Math.104 (2008), 47–81], who considered the setting of locally compact separable metric spaces. Our extension to Polish spaces solves an open problem posed by Zaharopol (loc. cit.) in a satisfactory manner.

In this paper, using the Kolmogorov–Arnold–Moser method we prove reducibility of a class of nonlinear quasi-periodic differential equation with degenerate equilibrium point under small perturbation and obtain a quasi-periodic solution near the equilibrium point.

Let X be a metrizable space with a continuous group or semi-group action G. Let D be a non-empty subset of X. Our problem is how to choose a fixed number of sets in {g−1D; g∈G}, say σ−1D with σ∈τ, to maximize the cardinality of the partition ℙ({σ−1D; σ∈τ}) generated by them. Let An infinite subset Σ of G is called an optimal position of the triple (X,G,D) if holds for any k=1,2,… and τ⊂Σ with #τ=k. In this paper, we discuss examples of the triple (X,G,D) admitting or not admitting an optimal position. Let X=G=ℝn  (n≥1) , where the action g∈G to x∈X is the translation x−g. If D is the n-dimensional unit ball, then holds and the triple (X,G,D) admits an optimal position. In fact, if n≥2 and Σ is an infinite subset of G such that for some δ with 0<δ<1 , Σ⊂{x∈ℝn ; ‖x‖=δ}, and that any subset of Σ with cardinality n+1 is not on a hyperplane, then Σ is an optimal position of the triple (X,G,D) . We have determined the primitive factor of the uniform sets coming from these optimal positions. We also show that in the above setting with n=2 and the unit square D′ in place of the unit disk D, the maximal pattern complexity is unchanged and p*X,G,D′(k)=k2 −k+2 , but there is no optimal position.