Description:
<jats:title>Abstract</jats:title><jats:p>We establish (some directions of) a <jats:italic>Ledrappier correspondence</jats:italic> between Hölder cocycles, Patterson–Sullivan measures, etc for word-hyperbolic groups with metric-Anosov Mineyev flow. We then study Patterson–Sullivan measures for <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385723000135_inline1.png" /><jats:tex-math>
$\vartheta $
</jats:tex-math></jats:alternatives></jats:inline-formula>-Anosov representations over a local field and show that these are parameterized by the <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385723000135_inline2.png" /><jats:tex-math>
$\vartheta $
</jats:tex-math></jats:alternatives></jats:inline-formula>-<jats:italic>critical hypersurface</jats:italic> of the representation. We use these Patterson–Sullivan measures to establish a dichotomy concerning directions in the interior of the <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385723000135_inline3.png" /><jats:tex-math>
$\vartheta $
</jats:tex-math></jats:alternatives></jats:inline-formula>-limit cone of the representation in question: if <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385723000135_inline4.png" /><jats:tex-math>
${\mathsf {u}}$
</jats:tex-math></jats:alternatives></jats:inline-formula> is such a half-line, then the subset of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385723000135_inline5.png" /><jats:tex-math>
${\mathsf {u}}$
</jats:tex-math></jats:alternatives></jats:inline-formula>-<jats:italic>conical limit points</jats:italic> has either total mass if <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385723000135_inline6.png" /><jats:tex-math>
$|\vartheta |\leq 2$
</jats:tex-math></jats:alternatives></jats:inline-formula> or zero mass if <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385723000135_inline7.png" /><jats:tex-math>
$|\vartheta |\geq 4.$
</jats:tex-math></jats:alternatives></jats:inline-formula> The case <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385723000135_inline8.png" /><jats:tex-math>
$|\vartheta |=3$
</jats:tex-math></jats:alternatives></jats:inline-formula> remains unsettled.</jats:p>