On Bifurcations of Dynamical Systems · 1974-pde-10

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Question 1

Fix a compact smooth manifold $M$, let $r>5$ and $k>1$, and let $\operatorname{Diff}^{r}(M)$ have the $C^{r}$ topology. Let $\Delta$ be the Morse--Smale diffeomorphisms, and let $\Phi$ be the $C^{k}$-arc space (with its $C^{k}$ topology) of arcs $\xi:[0,1]\to\operatorname{Diff}^{r}(M)$ with $\xi(0)\in\Delta$ and first exit time $b=\inf\{t:\xi(t)\notin\Delta\}<1$. For $f\in\operatorname{Diff}^{r}(M)$ and $x\in M$, let $\alpha_f(x)$ and $\omega_f(x)$ be the accumulation sets of the backward and forward iterates of $x$, respectively, and set $L(f)=\overline{\bigcup_{x\in M}\alpha_f(x)}\cup\overline{\bigcup_{x\in M}\omega_f(x)}$. Let $\Omega(f)$ be the set of points $x$ for which every neighborhood $U$ meets $f^n(U)$ for some $n>0$.

Assume that $L(\xi(b))$ is finite and hyperbolic and contains a cycle of hyperbolic periodic orbits, meaning periodic orbits $\mathcal O(p_0),\ldots,\mathcal O(p_j)=\mathcal O(p_0)$ with $W^u(\mathcal O(p_i))\cap W^s(\mathcal O(p_{i+1}))\ne\varnothing$ for every $i$, and with stable-manifold dimensions not all equal. Determine whether, for a residual subset of such arcs $\xi$, there are structurally stable diffeomorphisms $\xi(t)$ with $t>b$ arbitrarily close to $b$ whose nonwandering sets $\Omega(\xi(t))$ are infinite and zero-dimensional.

Source reference: Sheldon E. Newhouse and Jacob Palis, "Cycles and Bifurcation Theory," Asterisque 31, Societe Mathematique de France, 1976, pp. 43-140, Section 6 (the proposed extension of Theorem (4.2) to non-equidimensional cycles), https://www.numdam.org/item/AST_1976__31__43_0/.

Question 2

Fix a compact smooth manifold $M$, let $r>5$ and $k>1$, and let $\operatorname{Diff}^{r}(M)$ have the $C^{r}$ topology. Let $\Delta$ be the Morse--Smale diffeomorphisms, and let $\Phi$ be the $C^{k}$-arc space (with its $C^{k}$ topology) of arcs $\xi:[0,1]\to\operatorname{Diff}^{r}(M)$ with $\xi(0)\in\Delta$ and first exit time $b=\inf\{t:\xi(t)\notin\Delta\}<1$. For $f\in\operatorname{Diff}^{r}(M)$ and $x\in M$, let $\alpha_f(x)$ and $\omega_f(x)$ be the accumulation sets of the backward and forward iterates of $x$, respectively, and set $L(f)=\overline{\bigcup_{x\in M}\alpha_f(x)}\cup\overline{\bigcup_{x\in M}\omega_f(x)}$. A periodic orbit is quasi-hyperbolic if the derivative of its return map has exactly one simple neutral block (a simple eigenvalue $1$, a simple eigenvalue $-1$, or a simple complex-conjugate pair of modulus one) and all other multipliers are off the unit circle.

Assume that $L(\xi(b))$ is finite and that it contains a cycle of periodic orbits, meaning periodic orbits $\mathcal O(p_0),\ldots,\mathcal O(p_j)=\mathcal O(p_0)$ with $W^u(\mathcal O(p_i))\cap W^s(\mathcal O(p_{i+1}))\ne\varnothing$ for every $i$, and that one orbit in this cycle is quasi-hyperbolic. Determine whether, for a residual subset of such arcs $\xi$, there are structurally stable diffeomorphisms $\xi(t)$ with $t>b$ arbitrarily close to $b$.

Source reference: Sheldon E. Newhouse and Jacob Palis, "Cycles and Bifurcation Theory," Asterisque 31, Societe Mathematique de France, 1976, pp. 43-140, Section 6 (the proposed extension of Theorem (4.2) to a quasi-hyperbolic periodic point contained in a cycle), https://www.numdam.org/item/AST_1976__31__43_0/.