Lipschitz sub-actions for locally maximal hyperbolic sets of a $ C^1 $ map

Figure 1.  A schematic description of the grid $ Q_i(j,k) $ for $ n = 5 $. The horizontal axis is the unstable direction attached at each $ q_i $, the vertical axis is the stable direction. The dashed "horizontal lines" are obtained by iteration of the horizontal axes by the forward graph transform; they are graphs of small slope $ \alpha $. We highlight the positions of the two points $ q_i $ and $ f_{i-1}(q_{i-1}) $ at each index $ i $ to show that they must be close. The points $ Q_i(0,k) $, $ k \in \left[\kern-0.15em\left[ 0, i \right]\kern-0.15em\right] $, are obtained by intersecting the vertical axis with these dashed horizontal lines. The other points are obtained recursively, starting at $ i = n $, by taking the preimages by $ f_{i-1} $ of the dashed "vertically aligned" points at index $ i $ except those on the horizontal axis. These new points are pushed by $ f_{i-1}^{-1} $, down and to the right of the previously defined points $ Q_{i-1}(0,k) $. The representation as vertical dashed lines and the relative positions of the points $ Q_i(j,k) $ are only a convenient way to index the grid as a product $ (j,k) $ in $ \left[\kern-0.15em\left[ 0,n-i \right]\kern-0.15em\right] \times \left[\kern-0.15em\left[ 0, i \right]\kern-0.15em\right] $. The points $ p_i = Q_i(n-i,i) $ we are looking for are located at the upper right corner of the grid. By definition $ f_{i-1}(p_{i-1}) = p_i $

Figure 2.  The schematic $ r $ returns of Lemma 4.3. The orbit starts at time $ \tau_0 = 0 $ and waits until the last return time to the ball $ B_0 = B(x_{\tau_0},\epsilon_{AS}) $. Either there is no return time, then $ \tau_1 = 1 $; or $ x_n \in B_0 $, then $ r = 1 $ and $ \tau_1 = n $; or $ x_n \notin B_0 $, then $ \tau_1\geq2 $, $ \tau_1-1 $ is the last return and $ x_j \notin B_0 $ for every $ j \geq \tau_1 $. The orbit restarts at $ \tau_1 $, let $ B_1 = B(x_{\tau_1}, \epsilon_{AS}) $, wait until the last return to $ B_1 $, and so on