A general formulation for multi-branches complete chaotic maps that preserve a specified invariant density is provided and an implication relationship among this class of maps is revealed. Such relationship helps to derive a whole family of complete chaotic maps that preserve not only the same invariant measure but also the same degree of chaos in terms of Lyapunov component.

Much progress has been achieved in exploring the probabilistic characteristics of nonlinear dynamics from different aspects over the last ten years. While the most general formulations of unimodal (two-branches) complete chaotic maps that preserve general form of invariant densities have been provided in [

Let

A nonsingular map

there exists a partition

We will call

A complete map

ergodic with respect to the Lebesgue measure,

chaotic in the probabilistic sense, that is, an absolutely continuous invariant density

As illustrated in Figure

Illustration of branching function.

Depending on the sign indices of

if

if

In particular, we have

Instead of working on the branching function

A

Let

In particular, for

Therefore, for a complete chaotic map

A complete chaotic map

Upward and downward maps.

A complete map is called

A complete map is called

According to (

Let

Without loss of generality, we assume first

The “if” part follows directly from Corollary

To have (

The “only if” part can be seen more straightforward from the Frobenius-perron equation (

If

The assumption that

Therefore, we have

The facts of

The case in which

Due to the fact that

Two complete chaotic maps,

Figure

Mutually implied family.

By repeatedly applying Theorem

Two mutually implied complete chaotic maps preserve an identical invariant measure.

An important characteristic of the implication relationship is that it is invariant with smooth conjugations; that is, if

The implication relationship (the set of implication functions) is uniquely determined by the partition

Due to the uniqueness of implied family, just substituting (

A complete map

We further show that any member of an implication family shares the same degree of chaos.

All members of an implication family

Due to the facts that two conjugated complete maps have an identical Lyapunov exponent and that the implication relationship is invariant of topological conjugation, we can proceed with the complete maps that preserve the uniform invariant density (

For a complete map

It is thus sufficient to prove that the identity

Indeed, since

For

It is well known that

For the special case in which

Finally, we just point out that (

Consider a 3-to-1 complete chaotic map with

With

For this simple generating functions, we can apply directly formulation (

Moreover,

The three branching functions are given by

It is interesting to verify that

The above conclusions are illustrated in Figure

Illustration of Example

Invariant measure

Branching functions

Consider a 3-to-1 complete chaotic map with

With

The branching functions turn out to be

(i) Consider first the set of generating functions,

(ii) Alternatively, if we consider the set of generation functions

Accordingly, we have

Straight calculation reveals that

The above conclusions are illustrated in Figure

Illustration of Example

Invariant measure

Branching functions