We propose a new classification of periodic orbits of interval maps via over-rotation pairs. We prove a theorem for them similar to the Sharkovskii theorem.

We investigate different notions of recognizability for a free monoid morphism $\unicode[STIX]{x1D70E}:{\mathcal{A}}^{\ast }\rightarrow {\mathcal{B}}^{\ast }$. Full recognizability occurs when each (aperiodic) point in ${\mathcal{B}}^{\mathbb{Z}}$ admits at most one tiling with words $\unicode[STIX]{x1D70E}(a)$, $a\in {\mathcal{A}}$. This is stronger than the classical notion of recognizability of a substitution $\unicode[STIX]{x1D70E}:{\mathcal{A}}^{\ast }\rightarrow {\mathcal{A}}^{\ast }$, where the tiling must be compatible with the language of the substitution. We show that if $|{\mathcal{A}}|=2$, or if $\unicode[STIX]{x1D70E}$’s incidence matrix has rank $|{\mathcal{A}}|$, or if $\unicode[STIX]{x1D70E}$ is permutative, then $\unicode[STIX]{x1D70E}$ is fully recognizable. Next we investigate the classical notion of recognizability and improve earlier results of Mossé [Puissances de mots et reconnaissabilité des points fixes d’une substitution. Theoret. Comput. Sci.99(2) (1992), 327–334] and Bezuglyi et al [Aperiodic substitution systems and their Bratteli diagrams. Ergod. Th. & Dynam. Sys.29(1) (2009), 37–72], by showing that any substitution is recognizable for aperiodic points in its substitutive shift. Finally we define recognizability and also eventual recognizability for sequences of morphisms which define an $S$-adic shift. We prove that a sequence of morphisms on alphabets of bounded size, such that compositions of consecutive morphisms are growing on all letters, is eventually recognizable for aperiodic points. We provide examples of eventually recognizable, but not recognizable, sequences of morphisms, and sequences of morphisms which are not eventually recognizable. As an application, for a recognizable sequence of morphisms, we obtain an almost everywhere bijective correspondence between the $S$-adic shift it generates, and the measurable Bratteli–Vershik dynamical system that it defines.

In this paper we introduce foundational techniques and prove the following: if X is a \mathbb{Z}^d subshift without periodic points, if Y is a \mathbb{Z}^d square mixing subshift of finite type containing a finite orbit and if there exists a homomorphism X\rightarrow Y, then X embeds into Y if and only if h(X)<h(Y). For the proof, clopen markers are used to generate Voronoi tiles whose thickened boundaries are coded using the homomorphism. The entropy gap and the square mixing permit the construction of an injective code on the tile interiors. A second paper will show that \mathbb{Z}^2 square filling mixing shifts of finite type are square mixing and that homomorphisms exist, resulting in an extension of Krieger's Embedding Theorem to \mathbb{Z}^2 subshifts.

The aim of this paper is to show that the dynamics of Lp heat semigroups (p>2) on a symmetric space of non-compact type is very different from the dynamics of the Lp heat semigroups if 1<p≤2. To see this, we show that certain shifts of the Lp heat semigroups have a chaotic behavior if p>2, and that such a behavior is not possible in the cases 1<p≤2. These results are compared with the corresponding situation for Euclidean spaces and symmetric spaces of compact type, where such a behavior is not possible.

We investigate the ideal structures of the $C^*$-algebras arising from topological graphs. We give a complete description of ideals of such $C^*$-algebras that are invariant under the so-called gauge action, and give a condition on topological graphs so that all ideals are invariant under the gauge action. We get conditions for our $C^*$-algebras to be simple, prime or primitive. We completely determine the prime ideals, and show that most of them are primitive. Finally, we construct a discrete graph for which the associated $C^*$-algebra is prime but not primitive.

The existence of non-continuous invariant graphs (or strange non-chaotic attractors) in quasiperiodically forced systems has generated great interest, but there are still very few rigorous results about the properties of these objects. In particular, it is not known whether the topological closure of such graphs is typically a filled-in set, i.e. consists of a single interval on every fibre, or not. We give a positive answer to this question for the class of so-called pinched skew products, where non-continuous invariant graphs occur generically, provided that the rotation number on the base is diophantine and the system satisfies some additional conditions. For typical parameter families these conditions translate to a lower bound on the parameter. On the other hand, we also construct examples where the non-continuous invariant graphs contain isolated points, such that their topological closure cannot be filled-in.

We consider a piecewise continuous, piecewise monotone interval map and a piecewise constant weight. With these data we associate a weighted kneading matrix which generalizes the Milnor—Thurston matrix. We show that the determinant of this matrix is related to a natural weighted zeta function.

Working with well chosen Riemannian metrics and employing Nevanlinna’s theory, we make the thermodynamic formalism work for a wide class of hyperbolic meromorphic functions of finite order (including in particular exponential family, elliptic functions, cosine, tangent and the cosine–root family and also compositions of these functions with arbitrary polynomials). In particular, the existence of conformal (Gibbs) measures is established and then the existence of probability invariant measures equivalent to conformal measures is proven. As a geometric consequence of the developed thermodynamic formalism, a version of Bowen’s formula expressing the Hausdorff dimension of the radial Julia set as the zero of the pressure function and, moreover, the real analyticity of this dimension, is proved.

We study the convergence of N^{-1} \sum f_1(T^{a_1n}x)f_2(T^{a_2n}x)f_3(T^{a_3n}x), for a measure-preserving system (X, \mathcal{B}, \mu, T) and f_{1}, f_{2}, f_{3} \in L^{\infty}(\mu). This generalizes the theorem of Conze and Lesigne on such expressions and simplifies the proof. We also obtain a description of the limit.

Let M be a compact manifold that contains a two-dimensional punctured torus. Given p ∈ M and an integer r ≥ 2, there exists X ∈ ∞(M) having non-trivial recurrent trajectories and such that, for some neighbourhood of X|(M−{p}) in r(M−{p}), no Y ∈ has closed orbits.

We consider the problem of when does a positive entropy topological system have a continuous factor with strictly smaller entropy. In many cases it is shown that such small entropy factors exist. On the other hand, classes of examples are given where differentiable factors must preserve some of the original entropy.

We develop a Ruelle–Perron–Fröbenius transfer operator approach to the ergodic theory of a large class of non-uniformly expanding transformations on compact manifolds. For Hölder continuous potentials not too far from constant, we prove that the transfer operator has a positive eigenfunction, which is piecewise Hölder continuous, and use this fact to show that there is exactly one equilibrium state. Moreover, the equilibrium state is a non-lacunary Gibbs measure, a non-uniform version of the classical notion of Gibbs measure that we introduce here.

We study two classes of dynamical systems with holes: expanding maps of the interval and Collet–Eckmann maps with singularities. In both cases, we prove that there is a natural absolutely continuous conditionally invariant measure μ (a.c.c.i.m.) with the physical property that strictly positive Hölder continuous functions converge to the density of μ under the renormalized dynamics of the system. In addition, we construct an invariant measure ν, supported on the Cantor set of points that never escape from the system, that is ergodic and enjoys exponential decay of correlations for Hölder observables. We show that ν satisfies an equilibrium principle which implies that the escape rate formula, familiar to the thermodynamic formalism, holds outside the usual setting. In particular, it holds for Collet–Eckmann maps with holes, which are not uniformly hyperbolic and do not admit a finite Markov partition. We use a general framework of Young towers with holes and first prove results about the a.c.c.i.m. and the invariant measure on the tower. Then we show how to transfer results to the original dynamical system. This approach can be expected to generalize to other dynamical systems than the two above classes.

Here we prove that almost all interval exchange transformations which reverse orientation, in at least one interval, have a periodic point where the derivative is − 1. Therefore they are periodic in an open neighborhood of the periodic point.

This paper establishes a global dynamical systems approach for the fractal patterns which are obtained when analysing the divisibility of binomial coefficients modulo a prime power. The general framework is within the class of hierarchical iterated function systems. As a consequence we obtain a complete deciphering of the hierarchical self-similarity features.

We consider partially hyperbolic diffeomorphisms on compact manifolds. We define the notion of the unstable and stable foliations stably carrying some unique non-trivial homologies. Under this topological assumption, we prove the following two results: if the center foliation is one-dimensional, then the topological entropy is locally a constant; and if the center foliation is two-dimensional, then the topological entropy is continuous on the set of all diffeomorphisms. The proof uses a topological invariant we introduced, Yomdin’s theorem on upper semi-continuity, Katok’s theorem on lower semi-continuity for two-dimensional systems, and a refined Pesin–Ruelle inequality we proved for partially hyperbolic diffeomorphisms.

We study two properties of a finite measure-preserving dynamical system and a given sequence $({n}_{m} )$ of positive integers, namely rigidity and non-recurrence. Our goal is to find conditions on the sequence which ensure that it is, or is not, a rigid sequence or a non-recurrent sequence for some weakly mixing system or more generally for some ergodic system. The main focus is on weakly mixing systems. For example, we show that for any integer $a\geq 2$ the sequence ${n}_{m} = {a}^{m} $ is a sequence of rigidity for some weakly mixing system. We show the same for the sequence of denominators of the convergents in the continued fraction expansion of any irrational $\alpha $. We also consider the stronger property of IP-rigidity. We show that if $({n}_{m} )$ grows fast enough then there is a weakly mixing system which is IP-rigid along $({n}_{m} )$ and non-recurrent along $({n}_{m} + 1)$.

This work is concerned with the generic existence of homoclinic points for area preserving diffeomorphisms of compact orientable surfaces. We give a shorter proof of Pixton's theorem that shows that, Cr-generically, an area preserving diffeomorphism of the two sphere has the property that every hyperbolic periodic point has transverse homoclinic points. Then, we extend Pixton's result to the torus and investigate certain generic aspects of the accumulation of the invariant manifolds all over themselves in the case of symplectic diffeomorphisms of compact manifolds.

It is proved that special flows over irrational rotations and under functions whose Fourier coefficients are of order O(1/|n|) are disjoint in the sense of Furstenberg from all mixing flows. This is an essential strengthening of a classical result by Kočergin on the absence of mixing of special flows built over irrational rotations and under bounded variation roof functions.

We introduce and study a class of endomorphisms which are piecewise smooth and have hyperbolic attractors. We prove the existence of SBR measures and develop the stable manifold theory and ergodic theory of such maps. We show that the Young dimension formula holds if and only if the map is invertible SBR-almost everywhere.