Cherry Maps with Different Critical Exponents: Bifurcation of Geometry

Abstract

We consider order preserving $C^3$ circle maps with a flat piece, irrational rotation number and critical exponents $({\ell }_1,{\ell }_2)$. We detect a change in the geometry of the system. For $({\ell }_1,{\ell }_2)\in [1,2{]}^2$ the geometry is degenerate and it becomes bounded for $({\ell }_1,{\ell }_2)\in [2,\infty {)}^2\setminus \{(2,2)\}$. When the rotation number is of the form $[abab\cdots ]$; for some $a,b\in {\mathbb{N}}^{\ast }$, the geometry is bounded for $({\ell }_1,{\ell }_2)$ belonging above a curve defined on $]1,+\infty {[}^2$. As a consequence we estimate the Hausdorff dimension of the non-wandering set $K_f={\mathcal{S}}^1\setminus {\bigcup }_{i=0}^{\infty }{f}^{-i}(U)$. Precisely, the Hausdorff dimension of this set is equal to zero when the geometry is degenerate and it is strictly positive when the geometry is bounded.

arXiv comment23 pages. arXiv admin note: text overlap with arXiv:2103.02347

References in corpus (3)

• A Phase Transition for Circle Maps and Cherry Flows
• Invariant Manifolds for Non-differentiable Operators
• A Phase Transition for Circle Maps with a Flat Spot and Different Critical Exponents