This paper is concerned with distribution of maps with transversal homoclinic orbits in a continuous map space, which consists of continuous maps defined in a closed and bounded set of a Banach space. By the transversal homoclinic theorem, it is shown that the map space contains a dense set of maps that have transversal homoclinic orbits and are chaotic in the sense of both Li-Yorke and Devaney with positive topological entropy.

Distribution of a set of maps with some dynamical properties in some continuous map space is a very interesting topic. In the 1960s, Smale [

In 1963, Smale gave the well-known Smale-Birkhoff homoclinic theorem for diffeomorphisms [

Since 2004, Shi, Chen, and Yu extended the result about turbulent maps for one-dimensional maps introduced by Block and Coppel in 1992 [

In the present paper, we will construct a set of continuous chaotic maps with generalized transversal homoclinic orbits, and show that the set is dense in the continuous map space. The method used in the present paper is motivated by the idea in [

This paper is organized as follows. In Section

In this section, some notations and basic concepts are first introduced, including Li-Yorke and Devaney chaos, hyperbolic fixed point, and transversal homoclinic orbit. And then a useful lemma is given.

First, we give two definitions of chaos which will be used in the paper.

Let

Let

the periodic points of

Let

If

In the following, we first give the definition that two manifolds intersect transversally and then give the definition of transversal homoclinic orbit for continuous maps.

Two submanifolds

If

Let

An orbit

A homoclinic orbit

The following lemma is taken from Theorems 3.1, and 5.2, Corollary 6.1, and the result in Section 7 of [

Let

Let

Assume that

Let

Assume that

Note that it is not required that

In this section, we first consider distribution of maps with transversal homoclinic orbits in a continuous self-map space, which consists of continuous maps that transform a closed, bounded, and convex set in a Banach space into itself. At the end of this section, we discuss distribution of chaotic maps in a continuous map space, in which a map may not transform its domain into itself.

Without special illustration, we always assume that

In this section, we first study distribution of maps with transversal homoclinic orbits in

For convenience, by

For every map

For every map

Fix any

For any

Let

The rest of the proof is divided into three steps.

Construct a map

Define

For any

Next, define

Obviously,

From the definition of

Next, we will prove that

It follows from (

Set a positive constant

It is evident that

The entire proof is complete.

Let

the topological entropy

Let

Let

Set

By the discussions in Step

When it is not required that a map transforms its domain

Now, we only present the detailed result corresponding to Theorem

Let

the topological entropy

A general Banach space

(1) As we all know, under

(2) In the

This paper was supported by the RFDP of Higher Education of China (Grant 20100131110024) and the NNSF of China (Grant 11071143).