The Classification of Two Lines Mahavier Products

Abstract For non-negative real numbers r and $$\rho $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ρ</mml:mi> </mml:math> , with $$r&lt; 1 &lt; \rho $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>1</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mi>ρ</mml:mi> </mml:mrow> </mml:math> , let $$L_{r,\rho }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ρ</mml:mi> </mml:mrow> </mml:msub> </mml:math> be the union of two straight line segments in $$[0,1]\times [0,1]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>]</mml:mo> <mml:mo>×</mml:mo> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> with slopes r and $$\rho $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ρ</mml:mi> </mml:math> , one from (0, 0) to (1, r ) and the other from (0, 0) to $$(\frac{1}{\rho },0)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mi>ρ</mml:mi> </mml:mfrac> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . Recent work in Banič et al. (2024) and Gril Rogina (J Dyn Control Syst. 2023;29:1525–45) gives a partial characterization of the forward Mahavier products of such closed relations. But this work in Banič et al. (2024) and Gril Rogina (J Dyn Control Syst. 2023;29:1525–45) leaves open the problem of whether certain classes of these Mahavier products contain any pairwise homeomorphic continua. In this paper, we show that any two of the continua in the above mentioned classes are homeomorphic, thus completing the classification of these Mahavier products.