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2 votes
1 answer
187 views

How to find the hyperbolic dimension of map $f(z) = z^2$ of $\overline{\mathbb{C}}$ onto itself?

I am reading the research article "The Hausdorff dimension of the boundary of Mandelbrot set and Julia sets" by Shishikura. In his article, he defined hyperbolic sets and hyperbolic ...
Nirmal Rawat's user avatar
1 vote
1 answer
163 views

Existence of center-stable manifold when the Jacobian is singular?

The following is a result from Shub's monograph "Global Stability of Dynamical Systems". I dabble in the proof, and it appears to me that the existence of $W^{\rm cu}_{\rm loc}$ does not ...
Thomas's user avatar
  • 511
1 vote
0 answers
175 views

Example of topologically transitive dynamical system with invariant non-ergodic Borel measure

Let $U \subset M$ be an open subset of a Riemannian manifold. I’m trying to find or construct an example of a topologically transitive dynamical system $f : U \to U$ for which $f : \Lambda \to \...
D. Ford's user avatar
  • 151
2 votes
0 answers
124 views

On invariant cones of the Katok map

I am studying the Katok map and similarly constructed examples of nonuniformly hyperbolic surface diffeomorphisms. An important part of the analysis of these diffeomorphisms is the invariance of a ...
D. Ford's user avatar
  • 151