Stability is among the most important concepts in dynamical systems. Local stability is well-studied, whereas determining the ‘global stability’ of a nonlinear system is very challenging. Over the last few decades, many different ideas have been developed to address this issue, primarily driven by concrete applications. In particular, several disciplines suggested a web of concepts under the headline ‘resilience’. Unfortunately, there are many different variants and explanations of resilience, and often, the definitions are left relatively vague, sometimes even deliberately. Yet, to allow for a structural development of a mathematical theory of resilience that can be used across different areas, one has to ensure precise starting definitions and provide a mathematical comparison of different resilience measures. In this work, we provide a systematic review of the most relevant indicators of resilience in the context of continuous dynamical systems, grouped according to their mathematical features. The indicators are also generalised to be applicable to any attractor. These steps are important to ensure a more reliable, quantitatively comparable and reproducible study of resilience in dynamical systems. Furthermore, we also develop a new concept of resilience against certain nonautonomous perturbations to demonstrate how one can naturally extend our framework. All the indicators are finally compared via the analysis of a classic scalar model from population dynamics to show that direct quantitative application-based comparisons are an immediate consequence of a detailed mathematical analysis.

In this work, we explore the dynamical implications of a spectral sequence analysis of a filtered chain complex associated to a non-singular Morse–Smale (NMS) flow $\varphi $ on a closed orientable $3$-manifold $M^3$ with no heteroclinic trajectories connecting saddle periodic orbits. We introduce the novel concepts of cancellations and reductions of pairs of periodic orbits based on Franks’ morsification and Smale’s cancellation theorems. The main goal is to establish an algebraic-dynamical correspondence between the unfolding of this spectral sequence associated to $\varphi $ and a family of flows obtained by cancelling and reducing pairs of periodic orbits of $\varphi $ on $M^3$. This correspondence is achieved through a spectral sequence sweeping algorithm (SSSA), which determines the order in which these cancellations and reductions of periodic orbits occur, producing a family of NMS flows that reaches a core flow when the spectral sequence converges.

In this paper, we prove the continuity of iteration operators $\mathcal {J}_n$ on the space of all continuous self-maps of a locally compact Hausdorff space X and generally discuss dynamical behaviors of them. We characterize their fixed points and periodic points for $X=\mathbb {R}$ and the unit circle $S^1$. Then we indicate that all orbits of $\mathcal {J}_n$ are bounded; however, we prove that for $X=\mathbb {R}$ and $S^1$, every fixed point of $\mathcal {J}_n$ which is non-constant and equals the identity on its range is not Lyapunov stable. The boundedness and the instability exhibit the complexity of the system, but we show that the complicated behavior is not Devaney chaotic. We give a sufficient condition to classify the systems generated by iteration operators up to topological conjugacy.

Let $\alpha $ be a $C^{\infty }$ volume-preserving action on a closed n-manifold M by a lattice $\Gamma $ in $\mathrm {SL}(n,\mathbb {R})$, $n\ge 3$. Assume that there is an element $\gamma \in \Gamma $ such that $\alpha (\gamma )$ admits a dominated splitting. We prove that the manifold M is diffeomorphic to the torus ${{\mathbb T}^{n}={\mathbb R}^{n}/{\mathbb Z}^{n}}$ and $\alpha $ is smoothly conjugate to an affine action. Anosov diffeomorphisms and partial hyperbolic diffeomorphisms admit a dominated splitting. We obtained a topological global rigidity when $\alpha $ is $C^{1}$. We also prove similar theorems for actions on $2n$-manifolds by lattices in $\textrm {Sp}(2n,{\mathbb R})$ with $n\ge 2$ and $\mathrm {SO}(n,n)$ with $n\ge 5$.

We give an example of a path-wise connected open set of $C^{\infty }$ partially hyperbolic endomorphisms on the $2$-torus, on which the (unique) Sinai–Ruelle–Bowen (SRB) measure exists for each system and varies smoothly depending on the system, while the sign of its central Lyapunov exponent changes.

For a non-conformal repeller $\Lambda $ of a $C^{1+\alpha }$ map f preserving an ergodic measure $\mu $ of positive entropy, this paper shows that the Lyapunov dimension of $\mu $ can be approximated gradually by the Carathéodory singular dimension of a sequence of horseshoes. For a $C^{1+\alpha }$ diffeomorphism f preserving a hyperbolic ergodic measure $\mu $ of positive entropy, if $(f, \mu )$ has only two Lyapunov exponents $\unicode{x3bb} _u(\mu )>0>\unicode{x3bb} _s(\mu )$, then the Hausdorff or lower box or upper box dimension of $\mu $ can be approximated by the corresponding dimension of the horseshoes $\{\Lambda _n\}$. The same statement holds true if f is a $C^1$ diffeomorphism with a dominated Oseledet’s splitting with respect to $\mu $.

In this paper, we address the issue of synchronization of coupled systems, introducing concepts of local and global synchronization for a class of systems that extend the model of coupled map lattices. A criterion for local synchronization is given; numerical experiments are exhibited to illustrate the criteria and also to raise some questions in the end of the text.

The exponential ordering is exploited in the context of nonautonomous delay systems, inducing monotone skew-product semiflows under less restrictive conditions than usual. Some dynamical concepts linked to the order, such as semiequilibria, are considered for the exponential ordering, with implications for the determination of the presence of uniform persistence or the existence of global attractors. Also, some important conclusions on the long-term dynamics and attraction are obtained for monotone and sublinear delay systems for this ordering. The results are then applied to almost periodic Nicholson systems and new conditions are given for the existence of a unique almost periodic positive solution which asymptotically attracts every other positive solution.

In the context of discrete nonautonomous dynamics, we prove that the homeomorphisms in the linearization theorem are $C^2$ diffeomorphisms. In contrast to other related works, our result does not involve non-resonance conditions or spectral gaps. Our approach is based on the interlacing of the properties of nonautonomous hyperbolicity of the linear part, and boundedness and Lipschitzness of the nonlinearities. Moreover, we propose a functional approach to find conditions for regularity of arbitrary degree.

Let $({\mathbb X}, T)$ be a subshift of finite type equipped with the Gibbs measure $\nu $ and let f be a real-valued Hölder continuous function on ${\mathbb X}$ such that $\nu (f) = 0$. Consider the Birkhoff sums $S_n f = \sum _{k=0}^{n-1} f \circ T^{k}$, $n\geqslant 1$. For any $t \in {\mathbb R}$, denote by $\tau _t^f$ the first time when the sum $t+ S_n f$ leaves the positive half-line for some $n\geqslant 1$. By analogy with the case of random walks with independent and identically distributed increments, we study the asymptotic as $ n\to \infty $ of the probabilities $ \nu (x\in {\mathbb X}: \tau _t^f(x)>n) $ and $ {\nu (x\in {\mathbb X}: \tau _t^f(x)=n) }$. We also establish integral and local-type limit theorems for the sum $t+ S_n f(x)$ conditioned on the set $\{ x \in {\mathbb X}: \tau _t^f(x)>n \}.$

For a $C^1$ non-conformal repeller, this paper proves that there exists an ergodic measure of full Carathéodory singular dimension. For an average conformal hyperbolic set of a $C^1$ diffeomorphism, this paper constructs a Borel probability measure (with support strictly inside the repeller) of full Hausdorff dimension. If the average conformal hyperbolic set is of a $C^{1+\alpha }$ diffeomorphism, this paper shows that there exists an ergodic measure of maximal dimension.

We study local biholomorphisms with finite orbits in some neighborhood of the origin since they are intimately related to holomorphic foliations with closed leaves. We describe the structure of the set of periodic points in dimension 2. As a consequence we show that given a finite-orbits local biholomorphism F, in dimension 2, there exists an analytic curve passing through the origin and contained in the fixed-point set of some non-trivial iterate of $F.$ As an application we obtain that at least one eigenvalue of the linear part of F at the origin is a root of unity. Moreover, we show that such a result is sharp by exhibiting examples of finite-orbits local biholomorphisms such that exactly one of the eigenvalues is a root of unity. These examples are subtle since we show they cannot be embedded in one-parameter groups.

We show that a fibre-preserving self-diffeomorphism which has hyperbolic splittings along the fibres on a compact principal torus bundle is topologically conjugate to a map that is linear in the fibres.

We introduce a notion of sensitivity with respect to a continuous real-valued bounded map which provides a sufficient condition for a continuous transformation, acting on a Baire metric space, to exhibit a Baire generic subset of points with historic behavior (also known as irregular points). The applications of this criterion recover, and extend, several known theorems on the genericity of the irregular set, in addition to yielding a number of new results, including information on the irregular set of geodesic flows, in both negative and non-positive curvature, and semigroup actions.

In this paper, the turnpike property is established for a nonconvex optimal control problem in discrete time. The functional is defined by the notion of the ideal convergence and can be considered as an analogue of the terminal functional defined over infinite-time horizon. The turnpike property states that every optimal solution converges to some unique optimal stationary point in the sense of ideal convergence if the ideal is invariant under translations. This kind of convergence generalizes, for example, statistical convergence and convergence with respect to logarithmic density zero sets.

The manifold M is a Riemannian, boundaryless, and compact manifold with $\dim M\geq 2$, and f is a $C^{1+\beta }$ ($\kern0.3pt\beta>0$) diffeomorphism of M. $\varphi $ is a Hölder continuous potential on M. We construct an invariant and absolutely continuous family of measures (with transformation relations defined by $\varphi $), which sit on local unstable leaves. We present two main applications. First, given an ergodic homoclinic class $H_\chi (p)$, we prove that $\varphi $ admits a local equilibrium state on $H_\chi (p)$ if and only if $\varphi $ is ‘recurrent on $H_\chi (p)$’ (a condition tested by counting periodic points), and one of the leaf measures gives a positive measure to a set of positively recurrent hyperbolic points; and if an equilibrium measure exists, the said invariant and absolutely continuous family of measures constitute as its conditional measures. An immediate corollary is the local product structure of hyperbolic equilibrium states. Second, we prove a Ledrappier–Young property for hyperbolic equilibrium states: if $\varphi $ admits a conformal family of leaf measures and a hyperbolic local equilibrium state, then the leaf measures of the invariant family (respective to $\varphi $) are equivalent to the conformal measures (on a full measure set). This extends the celebrated result by Ledrappier and Young for hyperbolic Sinai–Ruelle–Bowen measures, which states that a hyperbolic equilibrium state of the geometric potential (with pressure zero) has conditional measures on local unstable leaves which are absolutely continuous with respect to the Riemannian volume of these leaves.

We discuss a metric description of the divergence of a (projectively) Anosov flow in dimension 3, in terms of its associated expansion rates and give metric and contact geometric characterizations of when a projectively Anosov flow is Anosov. We then study the symmetries that the existence of an invariant volume form yields on the geometry of an Anosov flow, from various viewpoints of the theory of contact hyperbolas, Reeb dynamics, and Liouville geometry, and give characterizations of when an Anosov flow is volume preserving, in terms of those theories. We finally use our study to show that the bi-contact surgery operations of Salmoiraghi [Surgery on Anosov flows using bi-contact geometry. Preprint, 2021, arXiv:2104.07109; and Goodman surgery and projectively Anosov flows. Preprint, 2022, arXiv:2202.01328] can be applied in an arbitrary small neighborhood of a periodic orbit of any Anosov flow. In particular, we conclude that the Goodman surgery of Anosov flows can be performed using the bi-contact surgery operations of Salmoiraghi [Goodman surgery and projectively Anosov flows. Preprint, 2022, arXiv:2202.01328].

Let $k\geq 2$ and $(X_{i}, \mathcal {T}_{i}), i=1,\ldots ,k$, be $\mathbb {Z}^{d}$-actions topological dynamical systems with $\mathcal {T}_i:=\{T_i^{\textbf {g}}:X_i{\rightarrow } X_i\}_{\textbf {g}\in \mathbb {Z}^{d}}$, where $d\in \mathbb {N}$ and $f\in C(X_{1})$. Assume that for each $1\leq i\leq k-1$, $(X_{i+1}, \mathcal {T}_{i+1})$ is a factor of $(X_{i}, \mathcal {T}_{i})$. In this paper, we introduce the weighted topological pressure $P^{\textbf {a}}(\mathcal {T}_{1},f)$ and weighted measure-theoretic entropy $h_{\mu }^{\textbf {a}}(\mathcal {T}_{1})$ for $\mathbb {Z}^{d}$-actions, and establish a weighted variational principle as

$$ \begin{align*} P^{\textbf{a}}(\mathcal{T}_{1},f)=\sup\bigg\{h_{\mu}^{\textbf{a}}(\mathcal{T}_{1})+\int_{X_{1}}f\,d\mu:\mu\in\mathcal{M}(X_{1}, \mathcal{T}_{1})\bigg\}. \end{align*} $$

This result not only generalizes some well-known variational principles about topological pressure for compact or non-compact sets, but also improves the variational principle for weighted topological pressure in [16] from $\mathbb {Z}_{+}$-action topological dynamical systems to $\mathbb {Z}^{d}$-actions topological dynamical systems.

For a pseudo-Anosov flow $\varphi $ without perfect fits on a closed $3$-manifold, Agol–Guéritaud produce a veering triangulation $\tau $ on the manifold M obtained by deleting the singular orbits of $\varphi $. We show that $\tau $ can be realized in M so that its 2-skeleton is positively transverse to $\varphi $, and that the combinatorially defined flow graph $\Phi $ embedded in M uniformly codes the orbits of $\varphi $ in a precise sense. Together with these facts, we use a modified version of the veering polynomial, previously introduced by the authors, to compute the growth rates of the closed orbits of $\varphi $ after cutting M along certain transverse surfaces, thereby generalizing the work of McMullen in the fibered setting. These results are new even in the case where the transverse surface represents a class in the boundary of a fibered cone of M. Our work can be used to study the flow $\varphi $ on the original closed manifold. Applications include counting growth rates of closed orbits after cutting along closed transverse surfaces, defining a continuous, convex entropy function on the ‘positive’ cone in $H^1$ of the cut-open manifold, and answering a question of Leininger about the closure of the set of all stretch factors arising as monodromies within a single fibered cone of a $3$-manifold. This last application connects to the study of endperiodic automorphisms of infinite-type surfaces and the growth rates of their periodic points.