What is the ergodic behaviour of numerically computed segments of orbits of a diffeomorphism? In this paper, we try to answer this question for a generic conservative $C^{1}$-diffeomorphism and segments of orbits of Baire-generic points. The numerical truncation is modelled by a spatial discretization. Our main result states that the uniform measures on the computed segments of orbits, starting from a generic point, accumulate on the whole set of measures that are invariant under the diffeomorphism. In particular, unlike what could be expected naively, such numerical experiments do not see the physical measures (or, more precisely, cannot distinguish physical measures from the other invariant measures).

We describe (infinite-dimensional) irreducible representations of the crossed product C$^{\ast }$-algebra associated with a topological dynamical system (based on $\mathbb{Z}$) and we show that their restrictions to the underlying $\ell ^{1}$-Banach $\ast$-algebra are not algebraically irreducible under mild conditions on the dynamical system. The above description of irreducible representations has two ingredients, ergodic measures on the space and ergodic extensions for the tensor product with type I factors, the latter of which may not have been explicitly taken up before but which will be explored by means of examples. A new class of ergodic measures is also constructed for irrational rotations on the circle.

We consider several different models for generating random fractals including random self-similar sets, random self-affine carpets, and Mandelbrot percolation. In each setting we compute either the almost sure or the Baire typical Assouad dimension and consider some illustrative examples. Our results reveal a phenomenon common to each of our models: the Assouad dimension of a randomly generated fractal is generically as big as possible and does not depend on the measure-theoretic or topological structure of the sample space. This is in stark contrast to the other commonly studied notions of dimension like the Hausdorff or packing dimension.

In this paper, we consider diffeomorphisms on a closed manifold $M$ preserving a hyperbolic Sinaĭ–Ruelle–Bowen probability measure $\unicode[STIX]{x1D707}$ having intersections for almost every pair of stable and unstable manifolds. In this context, we show the ergodicity of $\unicode[STIX]{x1D707}$ when the dimension of $M$ is at most three. If $\unicode[STIX]{x1D707}$ is smooth, then it is ergodic when the dimension of $M$ is at most four. As a byproduct of our arguments, we obtain sufficient (topological) conditions which guarantee that there exists at most one hyperbolic ergodic Sinaĭ–Ruelle–Bowen probability measure. Even in higher dimensional cases, we show that every transitive topological Anosov diffeomorphism admits at most one hyperbolic Sinaĭ–Ruelle–Bowen probability measure.

Extensive amenability is a property of group actions which has recently been used as a tool to prove amenability of groups. We study this property and prove that it is preserved under a very general construction of semidirect products. As an application, we establish the amenability of all subgroups of the group $\text{IET}$ of interval exchange transformations that have angular components of rational rank less than or equal to two. In addition, we obtain a reformulation of extensive amenability in terms of inverted orbits and use it to present a purely probabilistic proof that recurrent actions are extensively amenable. Finally, we study the triviality of the Poisson boundary for random walks on $\text{IET}$ and show that there are subgroups $G<\text{IET}$ admitting no finitely supported measure with trivial boundary.

The asymptotic behaviour of a cellular automaton iterated on a random configuration is well described by its limit probability measure(s). In this paper, we characterize measures and sets of measures that can be reached as limit points after iterating a cellular automaton on a simple initial measure. In addition to classical topological constraints, we exhibit necessary computational obstructions. With an additional hypothesis of connectivity, we show these computability conditions are sufficient by constructing a cellular automaton realizing these sets, using auxiliary states in order to perform computations. Adapting this construction, we obtain a similar characterization for the Cesàro mean convergence, a Rice theorem on the sets of limit points, and we are able to perform computation on the set of measures, i.e. the cellular automaton converges towards a set of limit points that depends on the initial measure. Last, under non-surjective hypotheses, it is possible to remove auxiliary states from the construction.

The joint spectral radius of a pair of $2\times 2$ real matrices $(A_{0},A_{1})\in M_{2}(\mathbb{R})^{2}$ is defined to be $r(A_{0},A_{1})=\limsup _{n\rightarrow \infty }\max \{\Vert A_{i_{1}}\cdots A_{i_{n}}\Vert ^{1/n}:i_{j}\in \{0,1\}\}$, the optimal growth rate of the norm of products of these matrices. The Lagarias–Wang finiteness conjecture [Lagarias and Wang. The finiteness conjecture for the generalized spectral radius of a set of matrices. Linear Algebra Appl.214 (1995), 17–42], asserting that $r(A_{0},A_{1})$ is always the $n$th root of the spectral radius of some length-$n$ product $A_{i_{1}}\cdots A_{i_{n}}$, has been refuted by Bousch and Mairesse [Asymptotic height optimization for topical IFS, Tetris heaps, and the finiteness conjecture. J. Amer. Math. Soc.15 (2002), 77–111], with subsequent counterexamples presented by Blondel et al [An elementary counterexample to the finiteness conjecture. SIAM J. Matrix Anal.24 (2003), 963–970], Kozyakin [A dynamical systems construction of a counterexample to the finiteness conjecture. Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference (Seville, Spain, December 2005). IEEE, Piscataway, NJ, pp. 2338–2343] and Hare et al [An explicit counterexample to the Lagarias–Wang finiteness conjecture. Adv. Math.226 (2011), 4667–4701]. In this article, we introduce a new approach to generating finiteness counterexamples, and use this to exhibit an open subset of $M_{2}(\mathbb{R})^{2}$ with the property that each member $(A_{0},A_{1})$ of the subset generates uncountably many counterexamples of the form $(A_{0},tA_{1})$. Our methods employ ergodic theory; in particular, the analysis of Sturmian invariant measures. This approach allows a short proof that the relationship between the parameter $t$ and the Sturmian parameter ${\mathcal{P}}(t)$ is a devil’s staircase.

We prove two generalizations of Furstenberg’s Diophantine result regarding the density of an orbit of an irrational point in the $1$-torus under the action of multiplication by a non-lacunary multiplicative semigroup of $\mathbb{N}$. We show that for any sequences $\{a_{n}\},\{b_{n}\}\subset \mathbb{Z}$ for which the quotients of successive elements tend to $1$ as $n$ goes to infinity, and any infinite sequence $\{c_{n}\}$, the set $\{a_{n}b_{m}c_{k}x:n,m,k\in \mathbb{N}\}$ is dense modulo $1$ for every irrational $x$. Moreover, by ergodic-theoretical methods, we prove that if $\{a_{n}\},\{b_{n}\}$ are a sequence having smooth $p$-adic interpolation for some prime number $p$, then for every irrational $x$, the sequence $\{p^{n}a_{m}b_{k}x:n,m,k\in \mathbb{N}\}$ is dense modulo 1.

We show that, on an oriented compact surface, two sufficiently $C^{2}$-close Riemannian metrics with strictly convex boundary, no conjugate points, hyperbolic trapped set for their geodesic flows and the same marked boundary distance are isometric via a diffeomorphism that fixes the boundary. We also prove that the same conclusion holds on a compact surface for any two negatively curved Riemannian metrics with strictly convex boundary and the same marked boundary distance, extending a result of Croke and Otal.

We study the interplay between the dynamics of area-preserving surface homeomorphisms homotopic to the identity and the topology of the surface. We define fully essential dynamics and generalize the results previously obtained on strictly toral dynamics to surfaces of higher genus. Non-fully essential dynamics are, in a way, reducible to surfaces of lower genus, while in the fully essential case the dynamics is decomposed into a disjoint union of periodic bounded disks and a complementary invariant externally transitive continuum $C$. When the Misiurewicz–Ziemian rotation set has non-empty interior the dynamics is fully essential, and the set $C$ is (externally) sensitive on initial conditions and realizes all of the rotational dynamics. As a fundamental tool we introduce the notion of homotopically bounded sets and we prove a general boundedness result for invariant open sets when the fixed point set is inessential.

Given an algebraic $\mathbf{Z}^{d}$-action corresponding to a prime ideal of a Laurent ring of polynomials in several variables, we show how to find the smallest order $n+1$ of non-mixing. It is known that this is determined by the non-mixing sets of size $n+1$, and we show how to find these in an effective way. When the underlying characteristic is positive and $n\geq 2$, we prove that there are at most finitely many classes under a natural equivalence relation. We work out two examples, the first with five classes and the second with 134 classes.

In this paper, we study Basmajian-type series identities on holomorphic families of Cantor sets associated to one-dimensional complex dynamical systems. We show that the series is absolutely summable if and only if the Hausdorff dimension of the Cantor set is strictly less than one. Throughout the domain of convergence, these identities can be analytically continued and they exhibit non-trivial monodromy.

In reversible dynamical systems, it is of great importance to understand symmetric features. The aim of this paper is to explore symmetric periodic points of reversible maps on planar domains invariant under a reflection. We extend Franks’ theorem on a dichotomy of the number of periodic points of area-preserving maps on the annulus to symmetric periodic points of area-preserving reversible maps. Interestingly, even a non-symmetric periodic point guarantees infinitely many symmetric periodic points. We prove an analogous statement for symmetric odd-periodic points of area-preserving reversible maps isotopic to the identity, which can be applied to dynamical systems with double symmetries. Our approach is simple, elementary, and far from Franks’ proof. We also show that a reversible map has a symmetric fixed point if and only if it is a twist map which generalizes a boundary twist condition on the closed annulus in the sense of Poincaré–Birkhoff. Applications to symmetric periodic orbits in reversible dynamical systems with two degrees of freedom are briefly discussed.

For non-invertible maps, subshifts that are mainly of finite type and piecewise monotone interval maps, we investigate what happens if we follow backward trajectories, which are random in the sense that, at each step, every preimage can be chosen with equal probability. In particular, we ask what happens if we try to compute the entropy this way. It turns out that, instead of the topological entropy, we get the metric entropy of a special measure, which we call the fair measure. In general, this entropy (the fair entropy) is smaller than the topological entropy. In such a way, for the systems that we consider, we get a new natural measure and a new invariant of topological conjugacy.

We prove that special flows over an ergodic rotation of the circle under a $C^{1}$ roof function with one discontinuity do not have local rank one. In particular, any such flow has infinite rank.

We consider interval exchange transformations of $n$ intervals with $k$ flips, or $(n,k)$-IETs for short, for positive integers $k,n$ with $k\leq n$. Our main result establishes the existence of minimal uniquely ergodic $(n,k)$-IETs when $n\geq 4$; moreover, these IETs are self-induced for $2\leq k\leq n-1$. This result extends the work on transitivity in Gutierrez et al [Transitive circle exchange transformations with flips. Discrete Contin. Dyn. Syst. 26(1) (2010), 251–263]. In order to achieve our objective we make a direct construction; in particular, we use the Rauzy induction to build a periodic Rauzy graph whose associated matrix has a positive power. Then we use a result in the Perron–Frobenius theory [Pullman, A geometric approach to the theory of non-negative matrices. Linear Algebra Appl. 4 (1971) 297–312] which allows us to ensure the existence of these minimal self-induced and uniquely ergodic $(n,k)$-IETs, $2\leq k\leq n-1$. We then find other permutations in the same Rauzy class generating minimal uniquely ergodic $(n,1)$- and $(n,n)$-IETs.

We consider commuting pairs of holomorphic endomorphisms of $\mathbb{P}^{2}$ with disjoint sequence of iterates. The case that has not been completely studied is when their degrees coincide after some number of iterations. We show in this case that they are either commuting Lattès maps or commuting homogeneous polynomial maps of $\mathbb{C}^{2}$ inducing a Lattès map on the line at infinity.

We study multi-sensitivity and thick sensitivity for continuous surjective selfmaps on compact metric spaces. Our main result states that a minimal system is either multi-sensitive or an almost one-to-one extension of its maximal equicontinuous factor. This is an analog of the Auslander–Yorke dichotomy theorem: a minimal system is either sensitive or equicontinuous. Furthermore, we introduce the concept of a syndetically equicontinuous point, and we prove that a transitive system is either thickly sensitive or contains syndetically equicontinuous points, which is a refinement of another well-known result of Akin, Auslander and Berg.

We prove a cocycle superrigidity theorem for a large class of coinduced actions. In particular, if $\unicode[STIX]{x1D6EC}$ is a subgroup of a countable group $\unicode[STIX]{x1D6E4}$, we consider a probability measure preserving action $\unicode[STIX]{x1D6EC}\curvearrowright X_{0}$ and let $\unicode[STIX]{x1D6E4}\curvearrowright X$ be the coinduced action. Assume either that $\unicode[STIX]{x1D6E4}$ has property (T) or that $\unicode[STIX]{x1D6EC}$ is amenable and $\unicode[STIX]{x1D6E4}$ is a product of non-amenable groups. Using Popa’s deformation/rigidity theory we prove $\unicode[STIX]{x1D6E4}\curvearrowright X$ is ${\mathcal{U}}_{\text{fin}}$-cocycle superrigid, that is any cocycle for this action to a ${\mathcal{U}}_{\text{fin}}$ (e.g. countable) group ${\mathcal{V}}$ is cohomologous to a homomorphism from $\unicode[STIX]{x1D6E4}$ to ${\mathcal{V}}.$

Consider an exact action of a discrete group $G$ on a separable C*-algebra $A$. It is shown that the reduced crossed product $A\rtimes _{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D706}}G$ is strongly purely infinite—provided that the action of $G$ on any quotient $A/I$ by a $G$-invariant closed ideal $I\neq A$ is element-wise properly outer and that the action of $G$ on $A$ is $G$-separating (cf. Definition 5.1). This is the first non-trivial sufficient general criterion for strong pure infiniteness of reduced crossed products of C*-algebras $A$ that are not $G$-simple. In the case $A=\text{C}_{0}(X)$, the notion of a $G$-separating action corresponds to the property that two compact sets $C_{1}$ and $C_{2}$, that are contained in open subsets $C_{j}\subseteq U_{j}\subseteq X$, can be mapped by elements $g_{1},g_{2}$ of $G$ onto disjoint sets $\unicode[STIX]{x1D70E}_{g_{j}}(C_{j})\subseteq U_{j}$, but satisfy not necessarily the contraction property $\unicode[STIX]{x1D70E}_{g_{j}}(U_{j})\subseteq \overline{U_{j}}$. A generalization of strong boundary actions on compact spaces to non-unital and non-commutative C*-algebras $A$ (cf. Definition 7.1) is also introduced. It is stronger than the notion of $G$-separating actions by Proposition 7.6, because $G$-separation does not imply $G$-simplicity and there are examples of $G$-separating actions with reduced crossed products that are stably projection-less and non-simple.