Typical Lipschitz images of rectifiable metric spaces

This article studies typical 1-Lipschitz images of 𝑛-rectifiable metric spaces 𝐸 into <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="double-struck">R</m:mi> <m:mi>m</m:mi> </m:msup> </m:math> Rᵐ for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>m</m:mi> <m:mo>≥</m:mo> <m:mi>n</m:mi> </m:mrow> </m:math> m≥ n . For example, if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>E</m:mi> <m:mo>⊂</m:mo> <m:msup> <m:mi mathvariant="double-struck">R</m:mi> <m:mi>k</m:mi> </m:msup> </m:mrow> </m:math> Eᵏ , we show that the Jacobian of such a typical 1-Lipschitz map equals 1 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="script">H</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> Hⁿ -almost everywhere and, if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>m</m:mi> <m:mo>></m:mo> <m:mi>n</m:mi> </m:mrow> </m:math> m>n , preserves the Hausdorff measure of 𝐸. In general, we provide sufficient conditions, in terms of the tangent norms of 𝐸, for when a typical 1-Lipschitz map preserves the Hausdorff measure of 𝐸, up to some constant multiple. Almost optimal results for strongly 𝑛-rectifiable metric spaces are obtained. On the other hand, for any norm <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mo>⋅</m:mo> <m:mo stretchy="false">|</m:mo> </m:mrow> </m:math> \,{·}\, on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="double-struck">R</m:mi> <m:mi>m</m:mi> </m:msup> </m:math> Rᵐ , we show that, in the space of 1-Lipschitz functions from <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mrow> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo stretchy="false">]</m:mo> </m:mrow> <m:mi>n</m:mi> </m:msup> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mo>⋅</m:mo> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mi mathvariant="normal">∞</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> ([-1,1]ⁿ,\,{·}\,∞) to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi mathvariant="double-struck">R</m:mi> <m:mi>m</m:mi> </m:msup> <m:mo>,</m:mo> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mo>⋅</m:mo> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (Rᵐ,\,{·}\,) , the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="script">H</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> Hⁿ -measure of a typical image is not bounded below by any <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="normal">Δ</m:mi> <m:mo>></m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> Δ>0 .