In this note, we examine the proportion of periodic orbits of Anosov flows that lie in an infinite zero density subset of the first homology group. We show that on a logarithmic scale we get convergence to a discrete fractal dimension.

Pavlov [Adv. Math. 295 (2016), 250–270; Nonlinearity 32 (2019), 2441–2466] studied the measures of maximal entropy for dynamical systems with weak versions of specification property and found the existence of intrinsic ergodicity would be influenced by the assumptions of the gap functions. Inspired by these, in this article, we study the dynamical systems with non-uniform specification property. We give some basic properties these systems have and give an assumption for the gap functions to ensure the systems have the following five properties: CO-measures are dense in invariant measures; for every non-empty compact connected subset of invariant measures, its saturated set is dense in the total space; ergodic measures are residual in invariant measures; ergodic measures are connected; and entropy-dense. In addition, we will give examples to show the assumption is optimal.

For every $r\in \mathbb {N}_{\geq 2}\cup \{\infty \}$, we prove a $C^r$-orbit connecting lemma for dynamically coherent and plaque expansive partially hyperbolic diffeomorphisms with one-dimensional orientation preserving center bundle. To be precise, for such a diffeomorphism f, if a point y is chain attainable from x through pseudo-orbits, then for any neighborhood U of x and any neighborhood V of y, there exist true orbits from U to V by arbitrarily $C^r$-small perturbations. As a consequence, we prove that for $C^r$-generic diffeomorphisms in this class, periodic points are dense in the chain recurrent set, and chain transitivity implies transitivity.

In this paper, we introduce the notion of planar two-center Stark–Zeeman systems and define four $J^{+}$ -like invariants for their periodic orbits. The construction is based on a previous construction for a planar one-center Stark–Zeeman system in [K. Cieliebak, U. Frauenfelder and O. van Koert. Periodic orbits in the restricted three-body problem and Arnold’s $J^+$ -invariant. Regul. Chaotic Dyn. 22(4) (2017), 408–434] as well as Levi-Civita and Birkhoff regularizations. We analyze the relationship among these invariants and show that they are largely independent, based on a new construction called interior connected sum.

We consider a convex Lagrangian $L:\mathit{TM}\rightarrow \mathbb{R}$ quadratic at infinity with $L(x,0)=0$ for every $x\in M$ and such that the 1-form $\unicode[STIX]{x1D703}$ defined by $\unicode[STIX]{x1D703}_{x}(v)=L_{v}(x,0)v$ is not closed. We show that for every number $a<0$, there is a contractible (nonconstant) periodic orbit with action $a$. We also obtain estimates of the period and energy of such periodic orbits.

We obtain an asymptotic formula for the number of pairs of closed orbits of a weak-mixing transitive Anosov flow whose homology classes have a fixed difference.