Let $M\stackrel {\rho _0}{\curvearrowleft }S$ be a $C^\infty $ locally free action of a connected simply connected solvable Lie group S on a closed manifold M. Roughly speaking, $\rho _0$ is parameter rigid if any $C^\infty $ locally free action of S on M having the same orbits as $\rho _0$ is $C^\infty $ conjugate to $\rho _0$ . In this paper we prove two types of result on parameter rigidity.

First let G be a connected semisimple Lie group with finite center of real rank at least $2$ without compact factors nor simple factors locally isomorphic to $\mathop {\mathrm {SO}}\nolimits _0(n,1)(n\,{\geq}\, 2)$ or $\mathop {\mathrm {SU}}\nolimits (n,1)(n\geq 2)$ , and let $\Gamma $ be an irreducible cocompact lattice in G. Let $G=KAN$ be an Iwasawa decomposition. We prove that the action $\Gamma \backslash G\curvearrowleft AN$ by right multiplication is parameter rigid. One of the three main ingredients of the proof is the rigidity theorems of Pansu, and Kleiner and Leeb on the quasi-isometries of Riemannian symmetric spaces of non-compact type.

Secondly we show that if $M\stackrel {\rho _0}{\curvearrowleft }S$ is parameter rigid, then the zeroth and first cohomology of the orbit foliation of $\rho _0$ with certain coefficients must vanish. This is a partial converse to the results in the author’s [Vanishing of cohomology and parameter rigidity of actions of solvable Lie groups. Geom. Topol. 21(1) (2017), 157–191], where we saw sufficient conditions for parameter rigidity in terms of vanishing of the first cohomology with various coefficients.

A diffeomorphism of the plane is Anosov if it has a hyperbolic splitting at every point of the plane. In addition to linear hyperbolic automorphisms, translations of the plane also carry an Anosov structure (the existence of Anosov structures for plane translations was originally shown by White). Mendes conjectured that these are the only topological conjugacy classes for Anosov diffeomorphisms in the plane. Very recently, Matsumoto gave an example of an Anosov diffeomorphism of the plane, which is a Brouwer translation but not topologically conjugate to a translation, disproving Mendes’ conjecture. In this paper we prove that Mendes’ claim holds when the Anosov diffeomorphism is the time-one map of a flow, via a theorem about foliations invariant under a time-one map. In particular, this shows that the kind of counterexample constructed by Matsumoto cannot be obtained from a flow on the plane.

We study the $L^{q}$ -spectrum of measures in the plane generated by certain nonlinear maps. In particular, we consider attractors of iterated function systems consisting of maps whose components are $C^{1+\alpha }$ and for which the Jacobian is a lower triangular matrix at every point subject to a natural domination condition on the entries. We calculate the $L^{q}$ -spectrum of Bernoulli measures supported on such sets by using an appropriately defined analogue of the singular value function and an appropriate pressure function.

We derive the almost sure Assouad spectrum and quasi-Assouad dimension of one-variable random self-affine Bedford–McMullen carpets. Previous work has revealed that the (related) Assouad dimension is not sufficiently sensitive to distinguish between subtle changes in the random model, since it tends to be almost surely ‘as large as possible’ (a deterministic quantity). This has been verified in conformal and non-conformal settings. In the conformal setting, the Assouad spectrum and quasi-Assouad dimension behave rather differently, tending to almost surely coincide with the upper box dimension. Here we investigate the non-conformal setting and find that the Assouad spectrum and quasi-Assouad dimension generally do not coincide with the box dimension or Assouad dimension. We provide examples highlighting the subtle differences between these notions. Our proofs combine deterministic covering techniques with suitably adapted Chernoff estimates and Borel–Cantelli-type arguments.

We study the rotation sets for homeomorphisms homotopic to the identity on the torus $\mathbb T^d$ , $d\ge 2$ . In the conservative setting, we prove that there exists a Baire residual subset of the set $\text {Homeo}_{0, \lambda }(\mathbb T^2)$ of conservative homeomorphisms homotopic to the identity so that the set of points with wild pointwise rotation set is a Baire residual subset in $\mathbb T^2$ , and that it carries full topological pressure and full metric mean dimension. Moreover, we prove that for every $d\ge 2$ the rotation set of $C^0$ -generic conservative homeomorphisms on $\mathbb T^d$ is convex. Related results are obtained in the case of dissipative homeomorphisms on tori. The previous results rely on the description of the topological complexity of the set of points with wild historic behavior and on the denseness of periodic measures for continuous maps with the gluing orbit property.

In 1932, von Neumann proposed classifying the statistical behavior of differentiable systems. Joint work of B. Weiss and the author proved that the classification problem is complete analytic. Based on techniques in that proof, one is able to show that the collection of recursive diffeomorphisms of the 2-torus that are isomorphic to their inverses is $\Pi ^0_1$-hard via a computable 1-1 reduction. As a corollary there is a diffeomorphism that is isomorphic to its inverse if and only if the Riemann Hypothesis holds, a different one that is isomorphic to its inverse if and only if Goldbach’s conjecture holds and so forth. Applying the reduction to the $\Pi ^0_1$-sentence expressing “ZFC is consistent” gives a diffeomorphism T of the 2-torus such that the question of whether $T\cong T^{-1}$ is independent of ZFC.

We describe topological obstructions (involving periodic points, topological entropy and rotation sets) for a homeomorphism on a compact manifold to embed in a continuous flow. We prove that homeomorphisms in a $C^{0}$-open and dense set of homeomorphisms isotopic to the identity in compact manifolds of dimension at least two are not the time-1 map of a continuous flow. Such property is also true for volume-preserving homeomorphisms in compact manifolds of dimension at least five. In the case of conservative homeomorphisms of the torus $\mathbb {T}^{d} (d\ge 2)$ isotopic to identity, we describe necessary conditions for a homeomorphism to be flowable in terms of the rotation sets.

This work investigates the existence and bifurcation structure of multi-pulse steady-state solutions to bistable lattice dynamical systems. Such solutions are characterized by multiple compact disconnected regions where the solution resembles one of the bistable states and resembles another trivial bistable state outside of these compact sets. It is shown that the bifurcation curves of these multi-pulse solutions lie along closed and bounded curves (isolas), even when single-pulse solutions lie along unbounded curves. These results are applied to a discrete Nagumo differential equation and we show that the hypotheses of this work can be confirmed analytically near the anti-continuum limit. Results are demonstrated with a number of numerical investigations.

Let $(A_m)_{m \in {\mathop Z}}$ be a sequence of bounded linear maps acting on an arbitrary Banach space X and admitting an exponential trichotomy and let $f_m:X \to X$ be a Lispchitz map for every $m\in {\mathop Z} $. We prove that whenever the Lipschitz constants of $f_m$, $m \in {\mathop Z} $, are uniformly small, the nonautonomous dynamics given by $x_{m+1}=A_mx_m+f_m(x_m)$, $m\in {\mathop Z} $, has various types of shadowing. Moreover, if X is finite dimensional and each $A_m$ is invertible we prove that a converse result is also true. Furthermore, we get similar results for one-sided and continuous time dynamics. As applications of our results, we study the Hyers–Ulam stability for certain difference equations and we obtain a very general version of the Grobman–Hartman's theorem for nonautonomous dynamics.

Consider an intermittent map $f_{\unicode[STIX]{x1D705}}:[0,1]\rightarrow [0,1]$ and a Hölder continuous potential $\unicode[STIX]{x1D711}:[0,1]\rightarrow \mathbb{R}$. We show that $\unicode[STIX]{x1D719}$ is stochastic for $f_{\unicode[STIX]{x1D705}}$ if and only if the topological pressure $P(f_{\unicode[STIX]{x1D705}},\unicode[STIX]{x1D711})$ satisfies $P(f_{\unicode[STIX]{x1D705}},\unicode[STIX]{x1D711})-\unicode[STIX]{x1D711}(0)>0$. As a consequence, for each $\unicode[STIX]{x1D6FD}>0$ sufficiently small, the set of Hölder continuous potentials of exponent $\unicode[STIX]{x1D6FD}$ that are not stochastic for $f_{\unicode[STIX]{x1D705}}$ has nonempty interior in the space of all such potentials.

Furstenberg [Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Syst. Theory1 (1967), 1–49] calculated the Hausdorff and Minkowski dimensions of one-sided subshifts in terms of topological entropy. We generalize this to $\mathbb{Z}^{2}$-subshifts. Our generalization involves mean dimension theory. We calculate the metric mean dimension and the mean Hausdorff dimension of $\mathbb{Z}^{2}$-subshifts with respect to a subaction of $\mathbb{Z}$. The resulting formula is quite analogous to Furstenberg’s theorem. We also calculate the rate distortion dimension of $\mathbb{Z}^{2}$-subshifts in terms of Kolmogorov–Sinai entropy.

Let $\operatorname{Homeo}_{+}(D_{n}^{2})$ be the group of orientation-preserving homeomorphisms of $D^{2}$ fixing the boundary pointwise and $n$ marked points as a set. The Nielsen realization problem for the braid group asks whether the natural projection $p_{n}:\operatorname{Homeo}_{+}(D_{n}^{2})\rightarrow B_{n}:=\unicode[STIX]{x1D70B}_{0}(\operatorname{Homeo}_{+}(D_{n}^{2}))$ has a section over subgroups of $B_{n}$. All of the previous methods use either torsion or Thurston stability, which do not apply to the pure braid group $PB_{n}$, the subgroup of $B_{n}$ that fixes $n$ marked points pointwise. In this paper, we show that the pure braid group has no realization inside the area-preserving homeomorphisms using rotation numbers.

Given a ${\mathcal{C}}^{\infty }$ expanding map $T$ of the circle, we construct a Hilbert space ${\mathcal{H}}$ of smooth functions on which the transfer operator ${\mathcal{L}}$ associated to $T$ acts as a compact operator. This result is made quantitative (in terms of singular values of the operator ${\mathcal{L}}$ acting on ${\mathcal{H}}$) using the language of Denjoy–Carleman classes. Moreover, the nuclear power decomposition of Baladi and Tsujii can be performed on the space ${\mathcal{H}}$, providing a bound on the growth of the dynamical determinant associated to ${\mathcal{L}}$.

We study Smale skew product endomorphisms (introduced in Mihailescu and Urbański [Skew product Smale endomorphisms over countable shifts of finite type. Ergod. Th. & Dynam. Sys. doi: 10.1017/etds.2019.31. Published online June 2019]) now over countable graph-directed Markov systems, and we prove the exact dimensionality of conditional measures in fibers, and then the global exact dimensionality of the equilibrium measure itself. Our results apply to large classes of systems and have many applications. They apply, for instance, to natural extensions of graph-directed Markov systems. Another application is to skew products over parabolic systems. We also give applications in ergodic number theory, for example to the continued fraction expansion, and the backward fraction expansion. In the end we obtain a general formula for the Hausdorff (and pointwise) dimension of equilibrium measures with respect to the induced maps of natural extensions ${\mathcal{T}}_{\unicode[STIX]{x1D6FD}}$ of $\unicode[STIX]{x1D6FD}$-maps $T_{\unicode[STIX]{x1D6FD}}$, for arbitrary $\unicode[STIX]{x1D6FD}>1$.

Let $\unicode[STIX]{x1D6E4}$ denote the mapping class group of the plane minus a Cantor set. We show that every action of $\unicode[STIX]{x1D6E4}$ on the circle is either trivial or semiconjugate to a unique minimal action on the so-called simple circle.

Kifer, Peres, and Weiss proved in [A dimension gap for continued fractions with independent digits. Israel J. Math.124 (2001), 61–76] that there exists $c_{0}>0$, such that $\dim \unicode[STIX]{x1D707}\leq 1-c_{0}$ for any probability measure $\unicode[STIX]{x1D707}$, which makes the digits of the continued fraction expansion independent and identically distributed random variables. In this paper we prove that amongst this class of measures, there exists one whose dimension is maximal. Our results also apply in the more general setting of countable branched systems.

Von Neumann’s original proof of the ergodic theorem is revisited. A uniform convergence rate is established under the assumption that one can control the density of the spectrum of the underlying self-adjoint operator when restricted to suitable subspaces. Explicit rates are obtained when the bound is polynomial, with applications to the linear Schrödinger and wave equations. In particular, decay estimates for time averages of solutions are shown.

We consider multimodal maps with holes and study the evolution of the open systems with respect to equilibrium states for both geometric and Hölder potentials. For small holes, we show that a large class of initial distributions share the same escape rate and converge to a unique absolutely continuous conditionally invariant measure; we also prove a variational principle connecting the escape rate to the pressure on the survivor set, with no conditions on the placement of the hole. Finally, introducing a weak condition on the centre of the hole, we prove scaling limits for the escape rate for holes centred at both periodic and non-periodic points, as the diameter of the hole goes to zero.

A Cantor action is a minimal equicontinuous action of a countably generated group $G$ on a Cantor space $X$. Such actions are also called generalized odometers in the literature. In this work, we introduce two new conjugacy invariants for Cantor actions, the stabilizer limit group and the centralizer limit group. An action is wild if the stabilizer limit group is an increasing sequence of stabilizer groups without bound and otherwise is said to be stable if this group chain is bounded. For Cantor actions by a finitely generated group $G$, we prove that stable actions satisfy a rigidity principle and furthermore show that the wild property is an invariant of the continuous orbit equivalence class of the action. A Cantor action is said to be dynamically wild if it is wild and the centralizer limit group is a proper subgroup of the stabilizer limit group. This property is also a conjugacy invariant and we show that a Cantor action with a non-Hausdorff element must be dynamically wild. We then give examples of wild Cantor actions with non-Hausdorff elements, using recursive methods from geometric group theory to define actions on the boundaries of trees.