Periodic orbits in the thin part of strata

Let S be a closed oriented surface of genus <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>g</m:mi> <m:mo>≥</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {g≥ 0} with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {n≥ 0} punctures and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:mrow> <m:mn>3</m:mn> <m:mo>⁢</m:mo> <m:mi>g</m:mi> </m:mrow> <m:mo>-</m:mo> <m:mn>3</m:mn> </m:mrow> <m:mo>+</m:mo> <m:mi>n</m:mi> </m:mrow> <m:mo>≥</m:mo> <m:mn>5</m:mn> </m:mrow> </m:math> {3g-3+n≥ 5} . Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒬</m:mi> </m:math> {{Q}} be a connected component of a stratum in the moduli space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">𝒬</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>S</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {{Q}(S)} of area one meromorphic quadratic differentials on S with n simple poles at the punctures or in the moduli space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">ℋ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>S</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {{H}(S)} of abelian differentials on S if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {n=0} . For a compact subset K of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">𝒬</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>S</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {{Q}(S)} or of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">ℋ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>S</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {{H}(S)} , we show that the asymptotic growth rate of the number of periodic orbits for the Teichmüller flow <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="normal">Φ</m:mi> <m:mi>t</m:mi> </m:msup> </m:math> {Φᵗ} on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒬</m:mi> </m:math> {{Q}} which are entirely contained in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">𝒬</m:mi> <m:mo>-</m:mo> <m:mi>K</m:mi> </m:mrow> </m:math> {{Q}-K} is at least <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>h</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="script">𝒬</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {h({Q})-1} , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>h</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="script">𝒬</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>></m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {h({Q})>0} is the complex dimension of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>ℝ</m:mi> <m:mo>+</m:mo> </m:msup> <m:mo>⁢</m:mo> <m:mi mathvariant="script">𝒬</m:mi> </m:mrow> </m:math> {R⁺{Q}} .