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This paper contains three new results. {\bf 1}.We introduce new notions ofprojective crystalline representations and twisted periodic Higgs-de Rhamflows. These new notions generalize crystalline representations of \'etalefundamental groups introduced in [7,10] and periodic Higgs-de Rham flowsintroduced in [19]. We establish an equivalence between the categories ofprojective crystalline representations and twisted periodic Higgs-de Rham flowsvia the category of twisted Fontaine-Faltings module which is also introducedin this paper. {\bf 2.}We study the base change of these objects over veryramified valuation rings and show that a stable periodic Higgs bundle givesrise to a geometrically absolutely irreducible crystalline representation. {\bf3.} We investigate the dynamic of self-maps induced by the Higgs-de Rham flowon the moduli spaces of rank-2 stable Higgs bundles of degree 1 on$\mathbb{P}^1$ with logarithmic structure on marked points$D:=\{x_1,\,...,x_n\}$ for $n\geq 4$ and construct infinitely manygeometrically absolutely irreducible $\mathrm{PGL_2}(\mathbbZ_p^{\mathrm{ur}})$-crystalline representations of$\pi_1^\text{et}(\mathbb{P}^1_{{\mathbb{Q}}_p^\text{ur}}\setminus D)$. We findan explicit formula of the self-map for the case $\{0,\,1,\,\infty,\,\lambda\}$and conjecture that a Higgs bundle is periodic if and only if the zero of theHiggs field is the image of a torsion point in the associated elliptic curve$\mathcal{C}_\lambda$ defined by $ y^2=x(x-1)(x-\lambda)$ with the ordercoprime to $p$.

Projective Crystalline Representations of \'Etale Fundamental Groups and Twisted Periodic Higgs-de Rham Flow