Description:
<p>In this paper we prove that <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Z left-parenthesis f right-parenthesis equals StartSet f Superscript k Baseline vertical-bar k element-of upper Z EndSet">
<mml:semantics>
<mml:mrow>
<mml:mi>Z</mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>f</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo>=</mml:mo>
<mml:mo fence="false" stretchy="false">{</mml:mo>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:msup>
<mml:mi>f</mml:mi>
<mml:mi>k</mml:mi>
</mml:msup>
</mml:mrow>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mo stretchy="false">|</mml:mo>
</mml:mrow>
<mml:mi>k</mml:mi>
<mml:mo>∈<!-- ∈ --></mml:mo>
<mml:mi>Z</mml:mi>
<mml:mo fence="false" stretchy="false">}</mml:mo>
</mml:mrow>
<mml:annotation encoding="application/x-tex">Z(f) = \{ {f^k}|k \in Z\}</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> for generic Axiom A diffeomorphisms. We also prove that generic diffeomorphisms have no <italic>k</italic>-roots.</p>