Symmetry lines and periodic points in the baker's map (Q1011793)

The baker's map from the unit square \([0,1]\times [0,1]\) to itself, is defined by \[ T(x,y)= \begin{cases} (2x,y/2)\quad &\text{if }0\leq x< 1/2,\;0\leq y\leq 1,\\ (2x- 1,{y+1\over 2})\quad &\text{if }1/2\leq x\leq 1,\;0\leq y\leq 1\end{cases} \] in this paper. The authors' intention is to investigate the periodic points of the baker's map using what they call the method of symmetric lines. To do this, they note that the baker's map can be factored as \(T= I_1,I_0\) where \(I_1\) and \(I_0\) are involutions. This factorization holds every where on the unit square except for the boundary line segment \(y= 1\), \(0\leq x< 1/2\). (This does not influence calculation of periodic points since the only periodic points on the boundary are \((0,0)\) and \((1,1)\).) The symmetry line \(\Gamma_n\) is defined as the set of fixed points of the involution \(I_n= T^n I_0\). Any symmetry line can be obtained from \(\Gamma_0\), \(\Gamma_1\) and \(T\) using the properties \(T^n\Gamma_j= \Gamma_{j+2n}\) and \(I_0\Gamma_j= \Gamma_{-j}\). The involution \(S= (1-x, 1-y)\) is also of interest. The symmetry lines of the baker's map are invariant under \(S\). Using symmetry lines and families of involutions, the authors provide an algorithm for computing periodic points. They note that if \(y\neq x\) and \(y\neq 1-x\), then whenever \((x,y)\) is a point of perioid \(k\), then so are \((y,x)\), \((1- x,1-y)\) and \((1-y, 1-x)\), associated with the involutions \(I_0\), \(S\) and \(SI_0\) applied to \((x,y)\). (These are four points corresponding to the vertices of a rectangle centered at \((1/2, 1/2)\) having sides rotated by \(\pi/4\) with respect to the axes.) There are no theorems stated, but complete arguments and calculations are provided throughout.