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 $(l_1, l_2)$. We detect a change in the geometry of the system. For $(l_1, l_2) \in [1, 2]^2$ the geometry is degenerate and becomes bounded for $(l_1, l_2) \in [2, \infty)^2 \backslash \{(2, 2)\}$. When the rotation number is of the form $[abab \ldots]$; for some $a, b \in \mathbb{N}^*$, the geometry is bounded for $(l_1, l_2)$ belonging above a curve defined on $]1, +\infty[^2$. As a consequence, we estimate the Hausdorff dimension of the nonwandering set $K_f=\mathcal{S}^1\backslash \bigcup^\infty_{i=0}f^{-i}(U)$. 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.

A phase transition for circle maps with a flat spot and different critical exponents

“…Suppose now that C u (n) = O(1). Then, by recalling the expression for w n , see (17), by (24) and ( 22), there exists a constant K such that 1 K ≤ S 2,2n , S 3,2n , S 5,2n ≤ K. We use now (20) and we find that, the same kind of bounds hold also for S 2,2n−1 , S 3,2n−1 and S 5,2n−1 . Moreover, because of ( 14), also the sequence S 1,n is bounded from below and from above.…”

“…Remark 2. Observe that, because of properties (20) and ( 21) we can restrict our analysis to the even sequence. The asymptotical behavior of the odd sequence can be expressed in the asymptotic of the even sequence and vice versa.…”

“…Introduction. The dynamics of circle maps with a flat interval has been intensively explored in the past years, see [20,4,10,17,16,18,19,22]. Among the many reasons to study these maps, they appear as first return map of a special flow on the torus, called Cherry flow, see [8,11,12,13,15,14].…”