Unstable manifolds and Hölder structures associated with noninvertible maps
The author tries to generalize the theory of dynamical systems from diffeomorphisms to non-invertible smooth maps. Let \(f: M\to M\) be a smooth map of a compact Riemannian manifold \(M\) into itself. The nonwandering set \(\Omega= \Omega_f\) of \(f\) is defined as usual. Assume that \(f\) has Axiom A, i.e. the periodic points of \(f\) are dense in \(\Omega\) and we have a splitting of the tangent bundle over the \(\widehat\Omega\), defined below, into subbundles, one of which is \(E^s\), consisting of contracting directions for the derivative \(Df\) and the other \(E^u\), consisting of the expanding directions for \(Df\). For a compact metric space \((X,d)\) and a continuous map \(f: X\to X\) we define the natural extension \(\widehat X\) of \(X\) with respect to \(f\) as \(\widehat X= \{\widehat x\mid\widehat x= (x,x_{-1}, x_{-2},\dots)\), \(f(x_{-i})= x_{i+1}\,(i\geq 1)\}\). The shift map on \(\widehat X\) is \(\widehat f:\widehat X\to\widehat X\), defined by \(\widehat f(\widehat x)= (f(x), x,x_{-1},\dots)\) and the canonical projection map \(\pi:\widehat X\to X\) is given by \(\pi(\widehat x)= x\) for \(\widehat x= (x,x_{-1},\dots)\in\widehat X\). There exists a natural metric and \(\widehat X\), for every \(K> 1\), given by \(d_K(\widehat x,\widehat y)= d(x,y)+ d(x_{-1}, y_{-1})/K+ d(x_{-2}, y_{-2})/K^2+\cdots\) for \(\widehat x,\widehat y\in\widehat X\). A continuous map \(f: X\to X\) is said to be (topologically) transitive on \(X\) if for any nonempty open sets \(U\), \(V\) in \(X\) there exists \(n\in\mathbb{Z}\) such that \(f^n(U)\cap V\neq\emptyset\). If \(f: M\to M\) is an Axiom A map as above, by the spectral decomposition theorem \(\Omega_f\) can be partitoned into a finite number of \(f\)-invariant subsets, on which \(f\) is transitive. These sets are unique up to order and are called basic sets. We say that a basic set \(\Lambda\) is of saddle type if there are both stable and unstable directions on \(\Lambda\). Now, given a map \(f: M\to M\) with Axiom A and a basic set \(\Lambda\) of saddle type, there exist local stable (resp. unstable) manifolds at every point \(x\in\Lambda\) (resp. \(\widehat x\in\widehat\Lambda\)) as follows: \(W^s_\beta(f,x)= \{y\in M\mid d(f^nx, f^ny)< \beta\), \(n\geq 0\}\), \(W^u_\beta(f,\widehat x)= \{y\in M\mid\exists\, y= (y,y_{-1},\dots)\in \widehat\Lambda\) with \(d(y_{-i}, x_{-i})< \beta\) \((i\geq 0)\}\), where \(\widehat x\in\widehat\Lambda\) and \(\beta\) is a suffuciently small positive number. We call the Hausdorff dimension \(\delta^u(\widehat x,\beta)= \text{HD}(W^u_\beta(f,\widehat x)\cap\Lambda)\) the unstable dimension (of size \(\beta\)) at the prehistory \(\widehat x\in\widehat\Lambda\). We denote the derivative in the unstable direction \(Df|_{E_{\hat x}^u}\) by \(Df_u(\widehat x)\) for \(\widehat x\in\widehat X\) and define the real function \(\varphi^u\) on \(\widehat X\) by \(\varphi^4(\widehat y)= -\log|Df_u(\widehat y)|\) for \(\widehat y\in\widehat X\). Now, one of the main theorems in this paper is the following: Let \(f: M\to M\) be an Axiom A smooth map of a compact Riemannian manifold \(M\) into itself. Consider a basic set \(\Lambda\) of saddle type and a small positive number \(\beta\) such that all unstable manifolds \(W^u_\beta(f,\widehat x)\) are defined. Then the unstable dimension \(\delta^u(\widehat x,\beta)\) is equal to the unique zero \(t^u\) of the pressure function \(t\mapsto P_{\widehat f}(t\cdot\varphi^u)\), the pressure \(P_f\) being defined for a continuous map \(f: X\to X\) as a functional \(P_f: C(X)\to\overline R\), where \(C(X)\) is the space of continuous \(\mathbb{R}\)-valued functions on \(X\). In particular, the unstable dimension does not depend on \(x\) and \(\beta\). The author next considers a smooth map \(f: M\to M\) with the property that for a compact subset \(\Lambda\), \(f(\Lambda)= \Lambda\), \(f|_\Lambda\) is transitive and \(f\) is hyperbolic and has local product structure over \(\Lambda\) and considers a perturbation \(g\) of \(f\). The author proves that there is a small neighborhood \({\mathcal U}\) of \(C^1(M,M)\) and a continuous map \(\Phi:{\mathcal U}\to C^0(\widehat\Lambda, M)\) such that \(\Phi(g)\circ\widehat f= f\circ\Phi(g)\) with several nice properties.
scientific article; zbMATH DE number 5127230