For a given topological dynamical system $T:X\rightarrow X$ over a compact set $X$ with a metric $d$, the variational principle states that

$$\begin{eqnarray}\sup _{\unicode[STIX]{x1D707}}h_{\unicode[STIX]{x1D707}}(T)=h(T)=h_{d}(T),\end{eqnarray}$$

$h_{\unicode[STIX]{x1D707}}(T)$

$T$

$h_{d}(T)$

$h(T)$

$T$

$X$

$$\begin{eqnarray}\sup _{\unicode[STIX]{x1D707}}h_{\unicode[STIX]{x1D707}}(T)=h(T)=\min _{d}h_{d}(T),\end{eqnarray}$$

$X$

$T$

$T:V\rightarrow V$

$V$

Let $f:X\rightarrow X$ be a continuous map on a compact metric space, let $\unicode[STIX]{x1D714}_{f}$ be the collection of $\unicode[STIX]{x1D714}$-limit sets of $f$ and let $\mathit{ICT}(f)$ be the collection of closed internally chain transitive subsets. Provided that $f$ has shadowing, it is known that the closure of $\unicode[STIX]{x1D714}_{f}$ in the Hausdorff metric coincides with $\mathit{ICT}(f)$. In this paper, we prove that $\unicode[STIX]{x1D714}_{f}=\mathit{ICT}(f)$ if and only if $f$ satisfies Pilyugin’s notion of orbital limit shadowing. We also characterize those maps for which $\overline{\unicode[STIX]{x1D714}_{f}}=\mathit{ICT}(f)$ in terms of a variation of orbital shadowing.

Using methods from ergodic theory along with properties of the Furstenberg measure we obtain conditions under which certain classes of plane self-affine sets have Hausdorff or box-counting dimensions equal to their affinity dimension. We exhibit some new specific classes of self-affine sets for which these dimensions are equal.

We show that for every ergodic measure-preserving system $(X,{\mathcal{X}},\unicode[STIX]{x1D707},S,T)$ with commuting transformations $S$ and $T$, the average

$$\begin{eqnarray}\frac{1}{N^{3}}\mathop{\sum }_{i,j,k=0}^{N-1}f_{0}(S^{j}T^{k}x)f_{1}(S^{i+j}T^{k}x)f_{2}(S^{j}T^{i+k}x)\end{eqnarray}$$

$\unicode[STIX]{x1D707}$

$x\in X$

$N\rightarrow \infty$

$f_{0},f_{1},f_{2}\in L^{\infty }(\unicode[STIX]{x1D707})$

$(X,{\mathcal{X}},\unicode[STIX]{x1D707},S,T)$

$$\begin{eqnarray}\frac{1}{N}\mathop{\sum }_{i=0}^{N-1}f_{1}(S^{i}x)f_{2}(T^{i}x)\end{eqnarray}$$

$\unicode[STIX]{x1D707}$

$x\in X$

$N\rightarrow \infty$

$f_{1},f_{2}\in L^{\infty }(\unicode[STIX]{x1D707})$

We define an oscillating sequence, an important example of which is generated by the Möbius function in number theory. We also define a minimally mean attractable (MMA) flow and a minimally mean-L-stable (MMLS) flow. One of the main results is that any oscillating sequence is linearly disjoint from all MMA and MMLS flows. In particular, this confirms Sarnak’s conjecture for all MMA and MMLS flows. We provide several examples of flows that are MMA and MMLS. These examples include flows defined by all $p$ -adic polynomials of integral coefficients, all $p$ -adic rational maps with good reduction, all automorphisms of the $2$ -torus with zero topological entropy, all diagonalizable affine maps of the $2$ -torus with zero topological entropy, all Feigenbaum maps, and all orientation-preserving circle homeomorphisms.

In this article, we provide sufficient conditions on a self-similar interval exchange map, whose renormalization matrix has complex eigenvalues of modulus greater than one, for the existence of affine interval exchange maps with wandering intervals that are semi-conjugate with it. These conditions are based on the algebraic properties of the complex eigenvalues and the complex fractals built from the natural substitution emerging from self-similarity. We show that the cubic Arnoux–Yoccoz interval exchange map satisfies these conditions.

We study the geometric and topological properties of strange non-chaotic attractors created in non-smooth saddle-node bifurcations of quasiperiodically forced interval maps. By interpreting the attractors as limit objects of the iterates of a continuous curve and controlling the geometry of the latter, we determine their Hausdorff and box-counting dimension and show that these take distinct values. Moreover, the same approach allows us to describe the topological structure of the attractors and to prove their minimality.

We construct a template with two ribbons that describes the topology of all periodic orbits of the geodesic flow on the unit tangent bundle to any sphere with three cone points with hyperbolic metric. The construction relies on the existence of a particular coding with two letters for the geodesics on these orbifolds.

We introduce two properties of dynamical systems on Polish metric spaces: closeability and linkability. We show that they imply density of ergodic measures in the space of invariant probability measures and the existence of a generic point for every invariant measure. In the compact case, it follows from our conditions that the set of invariant measures is either a singleton of a measure concentrated on a periodic orbit or the Poulsen simplex. We provide examples showing that closability and linkability are independent properties. Our theory applies to systems with the periodic specification property, irreducible Markov chains over a countable alphabet, certain coded systems including $\unicode[STIX]{x1D6FD}$-shifts and $S$-gap shifts, $C^{1}$-generic diffeomorphisms of a compact manifold $M$ and certain geodesic flows of a complete connected negatively curved manifold.

Given a self-map of a compact metric space $X$, we study periodic points of the map induced on the hyperspace of closed non-empty subsets of $X$. We give some necessary conditions on admissible sets of periods for these maps. Seemingly unrelated to this, we construct an almost totally minimal homeomorphism of the Cantor set. We also apply our theory to give a full description of admissible period sets for induced maps of the interval maps. The description of admissible periods is also given for maps induced on symmetric products.

For Hölder continuous cocycles over an invertible, Lipschitz base, we establish the Hölder continuity of Oseledets subspaces on compact sets of arbitrarily large measure. This extends a result of Araújo et al [On Hölder-continuity of Oseledets subspaces J. Lond. Math. Soc.93 (2016) 194–218] by considering possibly non-invertible cocycles, which, in addition, may take values in the space of compact operators on a Hilbert space. As a by-product of our work, we also show that a non-invertible cocycle with non-vanishing Lyapunov exponents exhibits non-uniformly hyperbolic behaviour (in the sense of Pesin) on a set of full measure.

The purpose of this paper is to consider the $N$-center problem with collinear centers, to identify its syzygy sequences that can be realized by minimizers of the Lagrangian action functional and to count the number of such syzygy sequences. In particular, we show that the number of such realizable syzygy sequences of length $\ell$ greater than or equal to two for the 3-center problem is at least $F_{\ell +2}-2$, where $\{F_{n}\}$ is the Fibonacci sequence. Moreover, with fixed length $\ell$, the density of such realizable syzygy sequences of length $\ell$ for the $N$-center problem approaches one as $N$ increases to infinity. Using reflection symmetry, the minimizers that we found can be extended to periodic solutions.

Let $\text{Homeo}_{0}(\mathbb{R}^{2};0)$ be the set of all homeomorphisms of the plane that are isotopic to the identity and which fix zero. Recently, in Le Roux [L’ensemble de rotation autour d’un point fixe. Astérisque (350) (2013), 1–109], Le Roux gave the definition of the local rotation set of an isotopy$I$ in $\text{Homeo}_{0}(\mathbb{R}^{2};0)$ from the identity to a homeomorphism $f$ and he asked if this set is always an interval. In this article, we give a positive answer to this question and to the analogous question in the case of the open annulus.

If $(T_{t})$ is a semigroup of Markov operators on an $L^{1}$-space that admits a non-trivial lower bound, then a well-known theorem of Lasota and Yorke asserts that the semigroup is strongly convergent as $t\rightarrow \infty$. In this article we generalize and improve this result in several respects. First, we give a new and very simple proof for the fact that the same conclusion also holds if the semigroup is merely assumed to be bounded instead of Markov. As a main result, we then prove a version of this theorem for semigroups which only admit certain individual lower bounds. Moreover, we generalize a theorem of Ding on semigroups of Frobenius–Perron operators. We also demonstrate how our results can be adapted to the setting of general Banach lattices and we give some counterexamples to show optimality of our results. Our methods combine some rather concrete estimates and approximation arguments with abstract functional analytical tools. One of these tools is a theorem which relates the convergence of a time-continuous operator semigroup to the convergence of embedded discrete semigroups.

We show how the fine structure in shift–tail equivalence, appearing in the non-commutative geometry of Cuntz–Krieger algebras developed by the first two listed authors, has an analogue in a wide range of other Cuntz–Pimsner algebras. To illustrate this structure, and where it appears, we produce an unbounded representative of the defining extension of the Cuntz–Pimsner algebra constructed from a finitely generated projective bi-Hilbertian module, extending work by the third listed author with Robertson and Sims. As an application, our construction yields new spectral triples for Cuntz and Cuntz–Krieger algebras and for Cuntz–Pimsner algebras associated to vector bundles twisted by an equicontinuous $\ast$-automorphism.

We construct a renewal structure for random walks on surface groups. The renewal times are defined as times when the random walks enter a particular type of cone and never leave it again. As a consequence, the trajectory of the random walk can be expressed as an aligned union of independent and identically distributed trajectories between the renewal times. Once having established this renewal structure, we prove a central limit theorem for the distance to the origin under exponential moment conditions. Analyticity of the speed and of the asymptotic variance are natural consequences of our approach. Furthermore, our method applies to groups with infinitely many ends and therefore generalizes classic results on central limit theorems on free groups.

We say that a countable group $G$ is McDuff if it admits a free ergodic probability measure preserving action such that the crossed product is a McDuff $\text{II}_{1}$ factor. Similarly, $G$ is said to be stable if it admits such an action with the orbit equivalence relation being stable. The McDuff property, stability, inner amenability and property Gamma are subtly related and several implications and non-implications were obtained in Effros [Property $\unicode[STIX]{x1D6E4}$ and inner amenability. Proc. Amer. Math. Soc.47 (1975), 483–486], Jones and Schmidt [Asymptotically invariant sequences and approximate finiteness. Amer. J. Math.109 (1987), 91–114], Vaes [An inner amenable group whose von Neumann algebra does not have property Gamma. Acta Math.208 (2012), 389–394], Kida [Inner amenable groups having no stable action. Geom. Dedicata173 (2014), 185–192] and Kida [Stability in orbit equivalence for Baumslag–Solitar groups and Vaes groups. Groups Geom. Dyn.9 (2015), 203–235]. We complete the picture with the remaining implications and counterexamples.

We show under weak hypotheses that $\unicode[STIX]{x2202}X$, the Roller boundary of a finite-dimensional CAT(0) cube complex $X$ is the Furstenberg–Poisson boundary of a sufficiently nice random walk on an acting group $\unicode[STIX]{x1D6E4}$. In particular, we show that if $\unicode[STIX]{x1D6E4}$ admits a non-elementary proper action on $X$, and $\unicode[STIX]{x1D707}$ is a generating probability measure of finite entropy and finite first logarithmic moment, then there is a $\unicode[STIX]{x1D707}$-stationary measure on $\unicode[STIX]{x2202}X$ making it the Furstenberg–Poisson boundary for the $\unicode[STIX]{x1D707}$-random walk on $\unicode[STIX]{x1D6E4}$. We also show that the support is contained in the closure of the regular points. Regular points exhibit strong contracting properties.

In this paper, we prove that if $R:\widehat{\mathbb{C}}\rightarrow \widehat{\mathbb{C}}$ is a postcritically finite rational map with Julia set homeomorphic to the Sierpiński carpet, then there is an integer $n_{0}$, such that, for any $n\geq n_{0}$, there exists an $R^{n}$-invariant Jordan curve $\unicode[STIX]{x1D6E4}$ containing the postcritical set of $R$.

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).