Hausdorff Dimension of Invariant Measures of Multicritical Circle Maps

Trujillo, Frank

doi:10.1007/s00023-020-00940-2

Cited by 5 publications

(3 citation statements)

References 34 publications

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“…Furthermore, each iterate is adjacent to the next one when seen as arcs in the circle; that is, they share a common endpoint. Geometric properties of these partitions are closely related to dimensional properties of the subjacent invariant measure, see, for example, [19,20,27].…”

Section: F Trujillomentioning

confidence: 99%

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On the Hausdorff dimension of invariant measures of piecewise smooth circle homeomorphisms

TRUJILLO

2024

Ergod. Th. Dynam. Sys.

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We show that, generically, the unique invariant measure of a sufficiently regular piecewise smooth circle homeomorphism with irrational rotation number and zero mean nonlinearity (e.g. piecewise linear) has zero Hausdorff dimension. To encode this generic condition, we consider piecewise smooth homeomorphisms as generalized interval exchange transformations (GIETs) of the interval and rely on the notion of combinatorial rotation number for GIETs, which can be seen as an extension of the classical notion of rotation number for circle homeomorphisms to the GIET setting.

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Section: F Trujillomentioning

confidence: 99%

“…If, in addition, the rotation number of the map is of bounded type, Graczyk and Świ ątek [15] showed that the Hausdorff dimension of the unique invariant measure is bounded away from 0 and 1. More recently, the author [27] provided explicit bounds, depending only on the arithmetic properties of the rotation number, for the Hausdorff dimension of these maps.…”

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confidence: 99%

On the Hausdorff dimension of invariant measures of piecewise smooth circle homeomorphisms

TRUJILLO

2024

Ergod. Th. Dynam. Sys.

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“…Therefore is just the pushforward of the Lebesgue measure under , that is, for any Borel set A in the unit circle (recall that the conjugacy h is unique up to post-composition with rotations, so the measure is well defined). In particular, has no atoms and gives positive measure to any open set (for more information on the measure , see [11, 12, 30] and references therein).…”

Section: Minimal Circle Homeomorphismsmentioning

confidence: 99%

Quasisymmetric orbit-flexibility of multicritical circle maps

Faria¹,

Guarino

2021

Ergod. Th. Dynam. Sys.

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Two given orbits of a minimal circle homeomorphism f are said to be geometrically equivalent if there exists a quasisymmetric circle homeomorphism identifying both orbits and commuting with f. By a well-known theorem due to Herman and Yoccoz, if f is a smooth diffeomorphism with Diophantine rotation number, then any two orbits are geometrically equivalent. It follows from the a priori bounds of Herman and Świątek, that the same holds if f is a critical circle map with rotation number of bounded type. By contrast, we prove in the present paper that if f is a critical circle map whose rotation number belongs to a certain full Lebesgue measure set in $(0,1)$ , then the number of equivalence classes is uncountable (Theorem 1.1). The proof of this result relies on the ergodicity of a two-dimensional skew product over the Gauss map. As a by-product of our techniques, we construct topological conjugacies between multicritical circle maps which are not quasisymmetric, and we show that this phenomenon is abundant, both from the topological and measure-theoretical viewpoints (Theorems 1.6 and 1.8).

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