We consider the group of foliation preserving Lipschitz homeomorphisms of a Lipschitz foliated manifold. First we show that the identity component of the group of leaf preserving Lipschitz homeomorphisms of a Lipschitz foliated manifold is perfect. Next using this result we compute the first homology of the group of foliation preserving Lipschitz homeomorphisms of a codimension one $C^{2}$-foliated manifold. Then we have results which are different from those of topological and differentiable cases.

In this paper we characterize the fiber representations of equivariant complex vector bundles over a circle and classify these bundles. We also treat the triviality of them by investigating the extensions of the fiber representations. As a corollary of our results, we calculate the reduced equivariant $K$-group of a circle for any compact Lie group, which extends a result of Y. Yang [Yan95].

There is a version of Lagrangian intersection theory in contact geometry [2]. But it works well only with very restrictive contact manifolds. For example, it does not work well with overtwisted contact 3-manifolds. Here we show the following. If we have an estimate on Hamiltonian functions of contact flow, then we can apply the theory to a much wider class of contact manifolds.

The family of polynomials $P_{n}: \mathbb{C}\times \mathbb{C} \to \mathbb{C}; (\lambda , z) \mapsto \lambda (1 + z/n)^{n}$ converges uniformly on compact subsets of the complex plane to the family of the complex exponentials $E:\mathbb{C}\times \mathbb{C} \to \mathbb{C}; (\lambda , z) \mapsto \lambda e^{z}$, as $n$ tends to infinity. Due to this convergence certain dynamical properties of the polynomials $P_{n}(\lambda , \cdot )$ carry over to the exponentials $E(\lambda , \cdot)$. Thus it possible to study entire transcendental maps, the exponentials, by considering polynomials for which the theory is well-known. Two particular problems have received attraction:

(1) For a fixed parameter $\lambda \in \mathbb{C}$ do the Julia sets of the polynomials $P_{n}(\lambda , \cdot )$ converge to the Julia set of $E(\lambda , \cdot)$?

(2) Do the hyperbolic components in the parameter space of $P_{n}$ converge to hyperbolic components of the family $E$?

In the present paper we study the Newton’s method associated with the entire transcendental functions $f(z) = p(z)e^{q(z)} + az + b$, with complex numbers $a$ and $b$, and complex polynomials $p$ and $q$. These functions $N_{f}$ can be approximated by the Newton’s method associated with $f_{m}(z) = p(z)(1+q(z)/m)^{m} +az +b$. In this paper we study the convergence of the Julia sets $\mathcal{J}(N_{f_{m}})\to \mathcal{J}(N_{f})$ and the Hausdorff convergence of hyperbolic components in the families $\{ N_{f_{m}}\}$ to the hyperbolic components of the family $\{ N_{f} \}$.