8/14/2019 On a P-Adic Julia Set

1/2

8/14/2019 On a P-Adic Julia Set

2/2

Communications of the Moscow Mathematical Society 1195

Proposition 3.6. All the finite periodic points of the map belong to K.

To prove this claim, we use Proposition 3.5 and the fact that the number of fixed points ofTn

does not exceed 2n.

Proposition 3.7. K coincides with the closure of the set of finite periodic points of the map T.

Proof. Let us observe that K is closed, as the intersection of closed sets of the form x+pkZp (see

the proof of Proposition 3.4). The fact that the set of p eriodic points ofT is dense in K follows

from the fact that the set of periodic sequences is dense in the set {1, 1}N

of all sequences.The question of the identity of the Julia set with the closure of the set of all repelling points

of the map was considered in [3]. It was conjectured there that the two sets are the same for any

rational map. Examples to support the conjecture were also given.

Proof of the theorem. Let us first prove assertion a).

The set K is invariant under T and T1. The closedness ofK follows from Proposition 3.7. Toprove minimality, it is enough to consider the restriction ofT to K. We use Proposition 3.5 and

note that we can consider only closed subsetsY of{1, 1}N that are invariant under the actionofL.Then clearly y Y k N (a0, a1, . . . , ak) {1, 1}

k+1 (Y (a0, a1, . . . , ak, y0, y1, y2, . . . )).

But then the fact that Y is closed implies that Y = {1, 1}N.Assertion b) follows now from Proposition 3.7.

Bibliography

[1] A. Yu. Khrennikov, Non-Archimedean analysis : quantum paradoxes, dynamical systems and

biological models, Kluwer, Dordrecht 1997.[2] N. Koblitz, p-Adic numbers, p-adic analysis, and zeta functions, 2nd ed., Springer-Verlag,

New YorkLondon 1984; Russian transl. of 1st ed., Mir, Moscow 1982.

[3] Liang-Chung Hsia, J. London Math. Soc. (2) 62 (2000), 685700.[4] R. Bowen, Methods of symbolic dynamics (collection of translations of Bowens papers), Mir,

Moscow 1979. (Russian)Received 23/SEP/03

http://dx.doi.org/10.1112/S0024610700001447http://dx.doi.org/10.1112/S0024610700001447http://dx.doi.org/10.1112/S0024610700001447http://dx.doi.org/10.1112/S0024610700001447http://dx.doi.org/10.1112/S0024610700001447http://dx.doi.org/10.1112/S0024610700001447http://dx.doi.org/10.1112/S0024610700001447http://dx.doi.org/10.1112/S0024610700001447http://dx.doi.org/10.1112/S0024610700001447http://dx.doi.org/10.1112/S0024610700001447