| Summary: | In this paper, we prove that on any contact manifold <inline-formula> <math display="inline"> <semantics> <mrow> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> </semantics> </math> </inline-formula> there exists an arbitrary <inline-formula> <math display="inline"> <semantics> <msup> <mi>C</mi> <mo>∞</mo> </msup> </semantics> </math> </inline-formula>-small contactomorphism which does not admit a square root. In particular, there exists an arbitrary <inline-formula> <math display="inline"> <semantics> <msup> <mi>C</mi> <mo>∞</mo> </msup> </semantics> </math> </inline-formula>-small contactomorphism which is not “autonomous„. This paper is the first step to study the topology of <inline-formula> <math display="inline"> <semantics> <mrow> <mi>C</mi> <mi>o</mi> <mi>n</mi> <msub> <mi>t</mi> <mn>0</mn> </msub> <mrow> <mrow> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> <mo>∖</mo> <mi>Aut</mi> <mrow> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </mrow> </semantics> </math> </inline-formula>. As an application, we also prove a similar result for the diffeomorphism group <inline-formula> <math display="inline"> <semantics> <mrow> <mi>Diff</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mrow> </semantics> </math> </inline-formula> for any smooth manifold <i>M</i>.
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