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Let  S  be a closed oriented surface of genusg≥0{g\geq 0}withn≥0{n\geq 0}punctures and3⁢g-3+n≥5{3g-3+n\geq 5}. Let{{\mathcal{Q}}}be a connected component of a stratum in the moduli space⁢(S){{\mathcal{Q}}(S)}of area one meromorphic quadratic differentials on  S  with  n  simple poles at the punctures or in the moduli spaceℋ⁢(S){{\mathcal{H}}(S)}of abelian differentials on  S  ifn=0{n=0}. For a compact subset  K  of⁢(S){{\mathcal{Q}}(S)}or ofℋ⁢(S){{\mathcal{H}}(S)}, we show that the asymptotic growth rate of the number of periodic orbits for the Teichmüller flowΦt{\Phi^{t}}on{{\mathcal{Q}}}which are entirely contained in-K{{\mathcal{Q}}-K}is at leasth⁢()-1{h({\mathcal{Q}})-1}, whereh⁢()&gt;0{h({\mathcal{Q}})&gt;0}is the complex dimension ofℝ+⁢{\mathbb{R}^{+}{\mathcal{Q}}}.

Periodic orbits in the thin part of strata