This is a little result I found during my work for graduation in Mathematics.
I'd like to know your opinion about it, so you can send me an e-mail to me@carlachi.com

You can also visit my new site: http://www.carlachi.com. Thank
you anyway.

**An application named sft**

Let** P **be the prime numbers set and let **P**(*i*) be the
*n*-th prime, **P**(1) = 2. Let *p* be a prime consisting of
*n* digits, we want to transform it in another *n*-digits prime. The
process is similar to that used for circular primes and it will be very clear
with an example: let *p* = 1997, now we shift *p*'s digits one
position left to obtain 997*u*, where *u* is an unknown digit.

We want 997*u *to be a prime so *u* must be in the set {1, 3, 7, 9}
and in our example the only possibility is *u* = 3.

In general we won't obtain a unique result, in fact for example starting from 1187 we'll have 1871, 1873, 1877 and 1879 which are all primes.

On the other hand there are primes, such as 8713, for which none of the four possible numbers is prime.

Let *p* be a prime consisting of four digits *abcd, *let's consider
the following set

then we'll erase the non prime elements from this set and we'll call
*N*(*p*) the resulting set.

We call *sft*(*p*) the application from **P **to the *elements
*of *N*(*p*).

In general, being *p* a *n*-digits prime, *N*(*p*) is
given erasing the non prime elements from the set

**Construction of the adjacency matrix and
evaluation of the entropy**

The adjacency matrix *C*[*n*] of this application for primes up to
*n*-digits is defined by the following rule: the (*i*, *j*)
element is 1 if we can go from **P**(*i*) to **P**(*j*) by the
application *sft*(**P**(*i*)), else the (*i*, *j*)
element is 0.

Computing the entropy of this matrix gives us an idea of the complexity of
the shift space resulting from the application *sft*.

Moreover entropy is an invariant (not complete) under conjugacy for this kind of spaces.

Using the Perron-Frobenius theory we can compute the entropy of the matrix
*C*[*n*], called *h*(*C*[*n*]), defined by the
following

- where L is the largest eigenvalue of
*C*[*n*].

This computation gives an unexpected result that is

the corrisponding Perron eigenvalue is L > 2.5611.

- This link will
show you the
*sft*application by a plot made starting from matrix*C*[4] in the following way: a point of coordinates (**P**(*i*),**P**(*j*)) is plotted corrisponding to the matrix elements equal to 1. - Moreover we can see the
*C*[4] matrix plotting a point to represent an element equal to 1. This two plots are strangely similar. **A stochastic dynamical system**

Iterating the application *sft* on the
prime number set **P **we build a dynamical system on this set. To have a
realization of such a system we must operate a choice everytime
*sft*(*p*) contains more then an element.

- If this choice is completely random we obtain a stochastic dynamical system as we can see from the following realizations: sometimes the orbits stop, for example

52792797

- because
*sft*(*p*) is empty; sometimes the orbits have no regularity and may continue indefinitely, for example

5279279179199199199799739739739339319319319119139137ecc.

If we state a way to choice inside the
*sft*(*p*) set we'll see a completely different behaviour of the
system: for example we can assume *min*(*sft*(*p*)) as iteration
of *p*, now we have a deterministic dynamical system .

We'll have orbits that stop after a while as

27417411411111171171

or more interesting orbits which become circular as

527927917919919919939931931131191193193193113119...

and

12232237237137197193193193113119119319319311...

In both the above cases the orbits become periodic as they reach the circular prime 9311. Moreover we can say that this two are the only possible behaviours for the points of such a dynamical system because it is a finite deterministic system.

The *sft *graphic contains every dynamical system that can be made by
the *C*[4] matrix.

The author is Carla Chicchiero under direction of Prof. P.E. Ricci, University "La Sapienza", Rome.

For the theory about shift spaces
see D. Lind and B. Marcus, *"An Introduction to Symbolic Dynamics and
Coding"*, Cambridge University Press.