Given positive integers $n$ and $m$, we consider dynamical systems in which (the disjoint union of) $n$ copies of a topological space is homeomorphic to $m$ copies of that same space. The universal such system is shown to arise naturally from the study of a C*-algebra denoted by ${\cal O}_{m,n}$, which in turn is obtained as a quotient of the well-known Leavitt C*-algebra $L_{m,n}$, a process meant to transform the generating set of partial isometries of $L_{m,n}$ into a tame set. Describing ${\cal O}_{m,n}$ as the crossed product of the universal $(m,n)$-dynamical system by a partial action of the free group $\mathbb {F}_{m+n}$, we show that ${\cal O}_{m,n}$ is not exact when $n$ and $m$ are both greater than or equal to 2, but the corresponding reduced crossed product, denoted by ${\cal O}_{m,n}^r$, is shown to be exact and non-nuclear. Still under the assumption that $m,n\geq 2$, we prove that the partial action of $\mathbb {F}_{m+n}$ is topologically free and that ${\cal O}_{m,n}^r$ satisfies property (SP) (small projections). We also show that ${\cal O}_{m,n}^r$admits no finite-dimensional representations. The techniques developed to treat this system include several new results pertaining to the theory of Fell bundles over discrete groups.

The structure of a self-similar set with the open set condition does not change under magnification. For self-affine sets, the situation is completely different. We consider self-affine Cantor sets $E\subset \mathbb {R}^2$ of the type studied by Bedford, McMullen, Gatzouras and Lalley, for which the projection onto the horizontal axis is an interval. We show that in small square $\varepsilon $-neighborhoods $N$ of almost each point $x$ in $E,$ with respect to many Bernoulli measures on the address space, $E\cap N$ is well approximated by product sets $[0,1]\times C$, where $C$ is a Cantor set. Even though $E$ is totally disconnected, all tangent sets have a product structure with interval fibers, reminiscent of the view of attractors of chaotic differentiable dynamical systems. We also prove that $E$has uniformly scaling scenery in the sense of Furstenberg, Gavish and Hochman: the family of tangent sets is the same at almost all points$x.$

Suppose $\sigma $ is the shift acting on Bernoulli space $X=\{0,1\}^{\mathbb {N}}$, and consider a fixed function $f:X \to \mathbb {R}$ satisfying the Walters conditions (defined in [P. Walters. A natural space of functions for the Ruelle operator theorem. Ergod. Th. & Dynam. Sys.27 (2007), 1323–1348]). For each real value $t\geq 0$ we consider the Ruelle operator $L_{\mathit {tf}}$. We are interested in the main eigenfunction $h_t$ of $L_{\mathit {tf}}$ and the main eigenmeasure $\nu _t$ for the dual operator $L_{\mathit {tf}}^*$, which we consider normalized in such a way that $h_t(0^\infty )=1$ and $\int h_t \,d\nu _t=1$ for all $t\gt 0$. We denote by $\mu _t= h_t \nu _t$ the Gibbs state for the potential $\mathit {tf}$. By the selection of a subaction $V$, when the temperature goes to zero (or $t\to \infty $), we mean the existence of the limit

\[ V:=\lim _{t\to \infty }\frac {1}{t}\log (h_{t}). \]

$\mu $

$t\to \infty $

\[\mu :=\lim _{t\to \infty } \mu _t.\]

$f$

$f$

$V$

We introduce topological pressure for continuous actions of countable sofic groups on compact metrizable spaces. This generalizes the classical topological pressure for continuous actions of countable amenable groups on such spaces. We also establish the variational principle for topological pressure in this sofic context.

Let $\Omega $ be a class of unital $C^*$-algebras. Then any simple unital $C^*$-algebra $A\in \mathrm {TA}(\mathrm {TA}\Omega )$ is a $\mathrm {TA}\Omega $ algebra. Let $A\in \mathrm {TA}\Omega $ be an infinite-dimensional $\alpha $-simple unital $C^*$-algebra with the property SP. Suppose that $\alpha :G\to \mathrm {Aut}(A)$ is an action of a finite group $G$ on $A$ which has a certain non-simple tracial Rokhlin property. Then the crossed product algebra $C^*(G,A,\alpha )$ belongs to $\mathrm {TA}\Omega $.

We describe a topological mechanism for the existence of diffusing orbits in a dynamical system satisfying the following assumptions: (i) the phase space contains a normally hyperbolic invariant manifold diffeomorphic to a two-dimensional annulus; (ii) the restriction of the dynamics to the annulus is an area preserving monotone twist map; (iii) the annulus contains sequences of invariant one-dimensional tori that form transition chains (i.e., the unstable manifold of each torus has a topologically transverse intersection with the stable manifold of the next torus in the sequence); (iv) the transition chains of tori are interspersed with gaps created by resonances; (v) within each gap there is prescribed a finite collection of Aubry–Mather sets. Under these assumptions, there exist trajectories that follow the transition chains, cross over the gaps, and follow the Aubry–Mather sets within each gap, in any specified order. This mechanism is related to the Arnold diffusion problem in Hamiltonian systems. In particular, we prove the existence of diffusing trajectories in the large gap problem of Hamiltonian systems. The argument is topological and constructive.

We extend the theory of transience to general dynamical systems with no Markov structure assumed. This is linked to the theory of phase transitions. We also provide new examples to illustrate different kinds of transient behaviour.

We study parameter families of quasi-periodically forced (qpf) circle maps with Diophantine frequency. Under certain $\mathcal {C}^1$-open conditions concerning their geometry, we prove that these families exhibit non-uniformly hyperbolic behaviour, often referred to as the existence of strange non-chaotic attractors, on parameter sets of positive measure. This provides a nonlinear version of results by Young on quasi-periodic $\mathrm {SL}(2,\mathbb {R})$-cocycles and complements previous results in this direction which hold for sets of frequencies of positive measure, but did not allow for an explicit characterization of these frequencies. As an application, we study a qpf version of the Arnold circle map and show that the Arnold tongue corresponding to rotation number $1/2$collapses on an open set of parameters. The proof requires to perform a parameter exclusion with respect to some twist parameter and is based on the multiscale analysis of the dynamics on certain dynamically defined critical sets. A crucial ingredient is to obtain good control on the parameter dependence of the critical sets. Apart from the presented results, we believe that this step will be important for obtaining further information on the behaviour of parameter families like the qpf Arnold circle map.

In this paper we will study families of circle maps of the form $x\mapsto x+ 2\pi r+ af(x)({\rm mod} \hspace{0.334em} 2\pi )$ and investigate how many periodic trajectories maps from this family can have for a ‘typical’ function $f$ provided the parameter $a$ is small.

In this paper we investigate how many periodic attractors maps in a small neighbourhood of a given map can have. For this purpose we develop new tools which help to make uniform cross-ratio distortion estimates in a neighbourhood of a map with degenerate critical points.

We prove that commutator subgroups of topological full groups arising from minimal subshifts have exponential growth. We also prove that the measurable full group associated to the countable, measure-preserving, ergodic and hyperfinite equivalence relation is topologically generated by two elements.

We compute the helicity of a vector field preserving a regular contact form on a closed three-dimensional manifold, and improve results of Gambaudo and Ghys [Enlacements asymptotiques. Topology 36(6) (1997), 1355–1379] relating the helicity of the suspension of a surface isotopy to the Calabi invariant of the isotopy. Based on these results, we provide positive answers to two questions posed by Arnold in [The asymptotic Hopf invariant and its applications. Selecta Math. Soviet. 5(4) (1986), 327–345]. In the presence of a regular contact form that is also preserved, the helicity extends to an invariant of an isotopy of volume-preserving homeomorphisms, and is invariant under conjugation by volume-preserving homeomorphisms. A similar statement also holds for suspensions of surface isotopies and surface diffeomorphisms. This requires the techniques of topological Hamiltonian and contact dynamics developed by Banyaga and Spaeth [On the uniqueness of generating Hamiltonians for topological strictly contact isotopies. Preprint, 2012], Buhovsky and Seyfaddini [Uniqueness of generating Hamiltonians for continuous Hamiltonian flows. J. Symplectic Geom. to appear, arXiv:1003.2612v2], Müller [The group of Hamiltonian homeomorphisms in the $L^\infty $-norm. J. Korean Math. Soc.45(6) (2008), 1769–1784], Müller and Oh [The group of Hamiltonian homeomorphisms and $C^0$-symplectic topology. J. Symplectic Geom. 5(2) (2007), 167–219], Müller and Spaeth [Topological contact dynamics I: symplectization and applications of the energy-capacity inequality. Preprint, 2011, arXiv:1110.6705v2] and Viterbo [On the uniqueness of generating Hamiltonian for continuous limits of Hamiltonians flows. Int. Math. Res. Not. (2006), 34028; Erratum, Int. Math. Res. Not. (2006), 38748]. Moreover, we generalize an example of Furstenberg [Strict ergodicity and transformation of the torus. Amer. J. Math. 83 (1961), 573–601] on topologically conjugate but not $C^1$-conjugate area-preserving diffeomorphisms of the two-torus to trivial $T^2$-bundles, and construct examples of Hamiltonian and contact vector fields that are topologically conjugate but not $C^1$-conjugate. Higher-dimensional helicities are considered briefly at the end of the paper.

Michael Handel proved in [A fixed-point theorem for planar homeomorphisms. Topology38 (1999), 235–264] the existence of a fixed point for an orientation-preserving homeomorphism of the open unit disk that can be extended to the closed disk, provided that it has points whose orbits form an oriented cycle of links at infinity. Later, Patrice Le Calvez gave a different proof of this theorem based only on Brouwer theory and plane topology arguments in [Une nouvelle preuve du théorème de point fixe de Handel. Geom. Topol.10(2006), 2299–2349]. These methods improved the result by proving the existence of a simple closed curve of index 1. We give a new, simpler proof of this improved version of the theorem and generalize it to non-oriented cycles of links at infinity.