Qualitative estimates of topological entropy for non-monotone contact Lax-Oleinik semiflow

For the non-monotone Hamilton-Jacobi equations of contact type, the associated Lax-Oleinik semiflow <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper T Subscript t Baseline comma upper C left-parenthesis upper M right-parenthesis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:mi>C</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(T_t, C(M))</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is expansive. In this paper, we provide qualitative estimates for both the lower and upper bounds of the topological entropy of the semiflow.