Solution of the 33rd Palis-Pugh problem for gradient-like diffeomorphisms of a two-dimensional sphere

Figure 1.  Conjugate arcs

Figure 2.  Saddle-node bifurcation

Figure 3.  Strongly stable and unstable foliations

Figure 4.  Flip bifurcation

Figure 5.  Illustration to the proof of lemma 3.1

Figure 6.  An example of an attractor $ A_f $ and repeller $ R_f $ for a gradient-like diffeomorphism $ f $

Figure 7.  For the shown diffeomorphism, there are two ways to choose the pair $ A_f,\,R_f $: 1) $ A_f = cl\,W^u_\sigma,\,R_f = \alpha $ and 2) $ A_f = \omega\cup f(\omega),\,R_f = cl\,W^s_\sigma $

Figure 8.  Illustration to the lemma 3.2

Figure 9.  Phase portrait of diffeomorphism $ \chi_m $

Figure 10.  Diffeomorphism $ \phi_{1, 3} $

Figure 11.  Illustration to the lemma 6.3, case 1)

Figure 12.  Illustration to the lemma 6.3, case 2)

Figure 13.  A diffeomorphism $ f\in G^+ $, for which $ m_f = 1 $, $ \mu_f = 3 $

Figure 14.  Transition from the diffeomorphism $ f\in G_m $ to the diffeomorphism $ g\in H_{m} $

Figure 15.  Curve $ C_\sigma $

Figure 16.  Curve $ C_{\tilde \sigma} $

Figure 17.  $ f\in H_{k,m} $

Figure 18.  $ f\in F_{1,3} $

Figure 19.  $ f \in {F_{1,3}}$

Figure 20.  ϕ_{1, 3}

Figure 21.  Illustration to the lemma 9.1, the case 1)

Figure 22.  Illustration to the lemma 9.1, case 1)

Figure 23.  Illustration to the lemma 9.1, case 2)