**Perron-Frobenius Operator**

Consider a map which possibly has a finite (or countable) number of discontinuities or points where possibly the derivative does not exist. We assume that there are points

or

such that restricted to each open interval is , with a bound on the first and the second derivatives. Assume that the interval ( or ) is positive invariant, so for all ( or for all ).

For such a map, we want a construction of a sequence of density functions that converge to a density function of an invariant measure. Starting with ( or ),assume that we have defined densities up to , then define define as follows

This operator , which takes one density function to another function, is called the Perron-Frobenius operator. The limit of the first density functions converges to a density function ,

The construction guarantees that is the density function for an invariant measure .

**Example 1.** Let

We construct the first few density functions by applying the Perron-Frobenius operator, which indicates the form of the invariant density function.

Take on . From the definition of , the slope on and are 1 and 2, respectively. If , then it has only one pre-image on ; else if , then it has two pre-images, one is in , the other one is in . Therefore,

By similar considerations,

By induction, we get

Now, we begin to calculate the density function . If , then

If , then

i.e.

**Example 2.** Let

Take on . By induction, on for all . Therefore, on .

**Example 3.** Let

Take on . Assume

for all . It is obviously that . By similar considerations,

That means

for all . From direct calculation, and for all . Therefore,

**Example 4.** Let

Take on . Assume

for all . It is obviously that . By similar considerations,

for all . From matrix diagonalization , and for all .

Therefore,

**Perron-Frobenius Theory**

**Definition.** Let be a matrix. We say is non-negative if for all . Such a matrix is called **irreducible** if for any pair there exists some such that where is the th element of . The matrix is** irreducible** and** aperiodic** if there exists such that for all .

**Perron-Frobenius Theorem** Let be a non-negative matrix.

(i) There is a non-negative eigenvalue such that no eigenvalue of has absolute value greater than .

(ii) We have .

(iii) Corresponding to the eigenvalue there is a non-negative left (row) eigenvector and a non-negative right (column) eigenvector .

(iv) If is irreducible then is a simple eigenvalue and the corresponding eigenvectors are strictly positive (i.e. , all ).

(v) If is irreducible then is the only eigenvalue of with a non-negative eigenvector.

**Theorem.**

Let be an irreducible and aperiodic non-negative matrix. Let and be the strictly positive eigenvectors corresponding to the largest eigenvalue as in the previous theorem. Then for each pair , .

Now, let us see previous examples, again. The matrix is irreducible and aperiodic non-negative matrix, and has the largest absolute value in the set of all eigenvalues of . From Perron-Frobenius Theorem, for all pairs . Then for each pari ,

. That means is a strictly positive matrix.

## Markov Maps

**Definition of Markov Maps. **Let be a compact interval. A map is called Markov if there exists a finite or countable family of disjoint open intervals in such that

(a) has Lebesgue measure zero and there exist and such that for each and each interval such that is contained in one of the intervals for each one has

(b) if , then ;

(c) there exists such that for each .

As usual, let be the Lebesgue measure on . We may assume that is a probability measure, i.e., . Usually, we will denote the Lebesgue measure of a Borel set by .

**Theorem. **Let be a Markov map and let be corresponding partition. Then there exists a invariant probability measure on the Borel sets of which is absolutely continuous with respect to the Lebesgue measure . This measure satisfies the following properties:

(a) its density is uniformly bounded and Holder continuous. Moreover, for each the density is either zero on or uniformly bounded away from zero.

If for every and one has for some then

(b) the measure is unique and its density is strictly positive;

(c) is exact with respect to ;

(d) for every Borel set .

If for each interval , then

(e) the density of is also uniformly bounded from below.