Julia sets and geometrically finite maps over finite extensions of the $p$-adic field

Abstract: Let

K

be a finite extension of the field

m a t h b b Q_{p}

of

p

-adic numbers, and

p h i i n K (z)

be a rational map of degree at least

2

. We prove that the

K

-Julia set of

p h i

is the natural restriction of

m a t h b b C_{p}

-Julia set, provided that the critical orbits are well-behaved. Moreover, under further assumption that

p h i

is geometrically finite, we prove that the dynamics on the

K

-Julia set of

p h i

is a countable state Markov shift.

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