Abstract:
We study a large class of suspension semiflows which contains the Lorenz semiflows. This is a class with low regularity (merely C^{1+\alpha}) and where the return map is discontinuous and the return time is unbounded. We establish the functional analytic framework which is typically employed to study rates of mixing. The Laplace transform of the correlation function is shown to admit a meromorphic extension to a strip about he imaginary axis. As part of this argument we give a new result concerning the quasi-compactness of weighted transfer operators for piecewise C^{1+\alpha} expanding interval maps.

Abstract:
We consider expanding semiflows on branched surfaces. The family of transfer operators associated to the semiflow is a one-parameter semigroup of operators. The transfer operators may also be viewed as an operator-valued function of time and so, in the appropriate norm, we may consider the vector-valued Laplace transform of this function. We obtain a spectral result on these operators and relate this to the spectrum of the generator of this semigroup. Issues of strong continuity of the semigroup are avoided. The main result is the improvement to the machinery associated with studying semiflows as one-parameter semigroups of operators and the study of the smoothness properties of semiflows defined on branched manifolds, without encoding as a suspension semiflow.

Abstract:
In this article we introduce a gluing orbit property, weaker than specification, for both maps and flows. We prove that flows with the $C^1$-robust gluing orbit property are uniformly hyperbolic and that every uniformly hyperbolic flow satisfies the gluing orbit property. We also prove a level-1 large deviations principle and a level-2 large deviations lower bound for for semiflows with the gluing orbit property. As a consequence we establish a level-1 large deviations principle for hyperbolic flows and every continuous observable, and also a level-2 large deviations lower bound. Finally, since many non-uniformly hyperbolic flows can be modeled as suspension flows we also provide criteria for such flows to satisfy uniform and non-uniform versions of the gluing orbit property.

Abstract:
Assuming some regularity of the dynamical zeta function, we establish an explicit formula with an error term for the prime orbit counting function of a suspended flow. We define the subclass of self-similar flows, for which we give an extensive analysis of the error term in the corresponding prime orbit theorem.

Abstract:
We obtain a exponential large deviation upper bound for continuous observables on suspension semiflows over a non-uniformly expanding base transformation with non-flat singularities or criticalities, where the roof function defining the suspension behaves like the logarithm of the distance to the singular/critical set of the base map. That is, given a continuous function we consider its space average with respect to a physical measure and compare this with the time averages along orbits of the semiflow, showing that the Lebesgue measure of the set of points whose time averages stay away from the space average tends to zero exponentially fast as time goes to infinity. Suspension semiflows model the dynamics of flows admitting cross-sections, where the dynamics of the base is given by the Poincar\'e return map and the roof function is the return time to the cross-section. The results are applicable in particular to semiflows modeling the geometric Lorenz attractors and the Lorenz flow, as well as other semiflows with multidimensional non-uniformly expanding base with non-flat singularities and/or criticalities under slow recurrence rate conditions to this singular/critical set.

Abstract:
We consider impulsive semiflows defined on compact metric spaces and give sufficient conditions, both on the semiflows and the potentials, for the existence and uniqueness of equilibrium states. We also generalize the classical notion of topological pressure to our setting of discontinuous semiflows and prove a variational principle.

Abstract:
In this paper we introduce a notion of an attractor for local semiflows on topological spaces, which in some cases seems to be more suitable than the existing ones in the literature. Based on this notion we develop a basic attractor theory on topological spaces under appropriate separation axioms. First, we discuss fundamental properties of attractors such as maximality and stability and establish some existence results. Then, we give a converse Lyapunov theorem. Finally, the Morse decomposition of attractors is also addressed.

Abstract:
In this article we prove a general theorem which establishes the existence of limiting distributions for a wide class of error terms from prime number theory. As a corollary to our main theorem, we deduce previous results of Wintner (1935), Rubinstein and Sarnak (1994), and of Ng (2004). In addition, we establish limiting distribution results for the error term in the prime number theorem for an automorphic $L$-function, weighted sums of the M\"{o}bius function, weighted sums of the Liouville function, the sum of the M\"{o}bius function in an arithmetic progression, and the error term in Chebotarev's density theorem.

Abstract:
We consider impulsive semiflows defined on compact metric spaces and deduce a variational principle. In particular, we generalize the classical notion of topological entropy to our setting of discontinuous semiflows.

Abstract:
In this paper we prove some linking theorems and mountain pass type results for dynamical systems in terms of local semiflows on complete metric spaces. Our results provide an alternative approach to detect the existence of compact invariant sets without using the Conley index theory. They can also be applied to variational problems of elliptic equations without verifying the classical P.S. Condition. As an example, we study the resonant problem of the nonautonomous parabolic equation $ u_t-\Delta u-\mu u=f(u)+g(x,t) $ on a bounded domain. The existence of a recurrent solution is proved under some Landesman-Laser type conditions by using an appropriate linking theorem of semiflows. Another example is the elliptic equation $-\Delta u+a(x)u=f(x,u)$ on $R^n$. We prove the existence of positive solutions by applying a mountain pass lemma of semiflows to the parabolic flow of the problem.