Local time and first return time for periodic semi-flows

Krickeberg mixing for Z extensions of Gibbs Markov semiflows

“…Let (Y, F, α, µ) be an ergodic measure preserving Gibbs Markov map. Let r : Y → R + be an L 1 (µ) roof function (called step time in [Th16]) and φ : Y → Z a displacement function (called step function in [Th16]). Throughout we assume that r is Lipschitz on each a ∈ α, and that φ is α-measurable with φ dµ = 0.…”

“…This is done via the present Theorem 1.1, which provides 'a smooth tail' estimate for the isomorphic semiflow (Ψ t ) t≥0 described below. The present arguments used in the proof of Theorem 1.1 build upon [Th16]. Given Theorem 1.1, the arguments required for the proof of Theorem 1.3 are essentially a 'translation' of the arguments in [G11] in the set up of [MT18].…”

“…As shown in [Th16], under certain assumptions on r and φ, the tail 1/µ(τ > t) is regularly varying with index less or equal to 1/2. To formulate our assumptions, for functions v that are Lipschitz on each a ∈ α, let |1 a v| ϑ = sup x =y∈a |v(x) − v(y)|/d ϑ (x, y), where d ϑ (x, y) = ϑ s(x,y) for some ϑ ∈ (0, 1) and s(x, y) = min{n : F n (x) and F n (y) are in different elements of α} is the separation time.…”

“…Next, we deal with the RHS using the following analogue of [Eri70, Lemma 7]: Before its statement, we briefly explain the strategy in [Th16] for obtaining the asymptotic of µ(τ > t) (such as (1.1)) and provide the main ingredients required in the statement and proof of Lemma 4.1. The key observation in [Th16] (also to be exploited here) is that the perturbed transfer operator R(u − ib) (associated with F ) can be understood via a double perturbation of the transfer operator for T , which we denote by L, perturbed with r and φ, respectively. For u, b ≥ 0 and θ…”

Multi-sensitivity, syndetical sensitivity and the asymptotic average- shadowing property for continuous semi-flows

“…In particular, they generalized the classical notion of topological entropy to their setting of discontinuous semi flows. In [49], Thomine studied Z d 1 -periodic semi-flows, which are versions in continuous time of Z d 1 -extensions of dynamical systems. These systems are defined by an underlying dynamical system, a step time (the time to wait before the system makes a move), and a step function (the displacement in Z d 1 at each step).…”