Hausdorff dimension of invariant measure of circle diffeomorphisms with a break point

Abstract

We prove that, for almost all irrational $\unicode[STIX]{x1D70C}\in (0,1)$, the Hausdorff dimension of the invariant measure of a $C^{2+\unicode[STIX]{x1D6FC}}$-smooth $(\unicode[STIX]{x1D6FC}\in (0,1))$ circle diffeomorphism with a break of size $c\in \mathbb{R}_{+}\backslash \{1\}$, with rotation number $\unicode[STIX]{x1D70C}$, is zero. This result cannot be extended to all irrational rotation numbers.

Hausdorff Dimension of Invariant Measures of Multicritical Circle Maps

“…Proof. Since ρ(f ) / ∈ D τ , there exists an increasing sequence (n k ) k∈N of natural numbers obeying (11) and define…”

“…In the case of circle homeomorphisms with a break, i.e. smooth diffeomorphisms with a singular point where the derivative has a jump discontinuity, Khanin and Kocić [11] have shown that for almost any irrational number α the unique invariant measure of a C 2+ circle homeomorphism with a break and rotation number equal to α has zero Hausdorff dimension. Ann.…”

“…In this work, we study the Hausdorff dimension of the unique invariant probability measures of certain piecewise smooth circle homeomorphisms, known as P -homeomorphisms or circle diffeomorphisms with breaks. We restrict ourselves to the setting of maps with zero mean nonlinearity, as the non-zero case has been recently treated by K. Khanin and S. Kocić in [17]. We refer the reader to Section 2.3 for the relevant definitions.…”

On the Hausdorff dimension of invariant measures for multicritical circle maps

“…Proof. Since ρ(f ) / ∈ D τ there exists an increasing sequence (n k ) k∈N of natural numbers obeying a n k +1 ≥ q τ n k for all k ∈ N. Given 0 < γ < 1, let A n i,γ , A n γ as in (11) and define…”

“…In the case of circle homeomorphisms with a break, i.e. smooth diffeomorphisms with a singular point where the derivative has a jump discontinuity, Khanin and Kocić [11] have shown that for almost any irrational number α the unique invariant measure of a C 2+ǫ circle homeomorphism with a break and rotation number equal to α has zero Hausdorff dimension.…”