Smooth, mixing transformations with loosely Bernoulli Cartesian square

A zero-entropy system is said to be loosely Bernoulli if it can be induced from an irrational rotation of the circle. We provide a criterion for zero-entropy systems to be loosely Bernoulli that is compatible with mixing. Using this criterion, we show the existence of smooth mixing zero-entropy loosely Bernoulli transformations whose Cartesian square is loosely Bernoulli.

A Note on the Conjugacy Between Two Critical Circle Maps

We study a conjugacy between two critical circle homeomorphisms with irrational rotation number. Let fi, i = 1, 2 be a C3 circle homeomorphisms with critical point x(i) cr of the order 2mi + 1. We prove that if 2m1 + 1 ̸= 2m2 + 1, then conjugating between f1 and f2 is a singular function. Keywords: circle homeomorphism, critical point, conjugating map, rotation number, singular function

Equicontinuous local dendrite maps

<p>Let X be a local dendrite, and f : X → X be a map. Denote by E(X) the set of endpoints of X. We show that if E(X) is countable, then the following are equivalent:</p><p>(1) f is equicontinuous;</p><p>(2) <img src="data:image/png;base64,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" alt="" /> fⁿ (X) = R(f);</p><p>(3) f| <img src="data:image/png;base64,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" alt="" /> fⁿ (X) is equicontinuous;</p><p>(4) f| <img src="data:image/png;base64,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" alt="" />fⁿ (X) is a pointwise periodic homeomorphism or is topologically conjugate to an irrational rotation of S 1 ;</p><p>(5) ω(x, f) = Ω(x, f) for all x ∈ X.</p><p>This result generalizes [17, Theorem 5.2], [24, Theorem 2] and [11, Theorem 2.8].</p>

Periodic point free homeomorphisms and irrational rotation factors

We provide a complete characterization of periodic point free homeomorphisms of the $2$ -torus admitting irrational circle rotations as topological factors. Given a homeomorphism of the $2$ -torus without periodic points and exhibiting uniformly bounded rotational deviations with respect to a rational direction, we show that annularity and the geometry of its non-wandering set are the only possible obstructions for the existence of an irrational circle rotation as topological factor. Through a very precise study of the dynamics of the induced $\rho $ -centralized skew-product, we extend and generalize considerably previous results of Jäger.

Punctured Parabolic Cylinders in Automorphisms of ℂ2

We show the existence of automorphisms $F$ of $\mathbb{C}^{2}$ with a non-recurrent Fatou component $\Omega $ biholomorphic to $\mathbb{C}\times \mathbb{C}^{*}$ that is the basin of attraction to an invariant entire curve on which $F$ acts as an irrational rotation. We further show that the biholomorphism $\Omega \to \mathbb{C}\times \mathbb{C}^{*}$ can be chosen such that it conjugates $F$ to a translation $(z,w)\mapsto (z+1,w)$, making $\Omega $ a parabolic cylinder as recently defined by L. Boc Thaler, F. Bracci, and H. Peters. $F$ and $\Omega $ are obtained by blowing up a fixed point of an automorphism of $\mathbb{C}^{2}$ with a Fatou component of the same biholomorphic type attracted to that fixed point, established by F. Bracci, J. Raissy, and B. Stensønes. A crucial step is the application of the density property of a suitable Lie algebra to show that the automorphism in their work can be chosen such that it fixes a coordinate axis. We can then remove the proper transform of that axis from the blow-up to obtain an $F$-stable subset of the blow-up that is biholomorphic to $\mathbb{C}^{2}$. Thus, we can interpret $F$ as an automorphism of $\mathbb{C}^{2}$.

Cherry Maps with Different Critical Exponents: Bifurcation of Geometry

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.

The K-inductive structure of the noncommutative Fourier transform

The noncommutative Fourier transform $\sigma (U)=V^{-1}$, $\sigma (V)=U$ of the irrational rotation C*-algebra $A_\theta $ (generated by canonical unitaries $U$, $V$ satisfying $VU = e^{2\pi i\theta } UV$) is shown to have the following K-inductive structure (for a concrete class of irrational parameters, containing dense $G_\delta $'s). There are approximately central matrix projections $e_1$, $e_2$, $f$ that are σ-invariant and which form a partition of unity in $K_0$ of the fixed-point orbifold $A_\theta ^\sigma $, where $f$ has the form $f = g+\sigma (g) +\sigma ^2(g)+\sigma ^3(g)$, and where $g$ is an approximately central matrix projection as well.

Möbius disjointness along ergodic sequences for uniquely ergodic actions

We show that there is an irrational rotation $Tx=x+\unicode[STIX]{x1D6FC}$ on the circle $\mathbb{T}$ and a continuous $\unicode[STIX]{x1D711}:\mathbb{T}\rightarrow \mathbb{R}$ such that for each (continuous) uniquely ergodic flow ${\mathcal{S}}=(S_{t})_{t\in \mathbb{R}}$ acting on a compact metric space $Y$, the automorphism $T_{\unicode[STIX]{x1D711},{\mathcal{S}}}$ acting on $(X\times Y,\unicode[STIX]{x1D707}\otimes \unicode[STIX]{x1D708})$ by the formula $T_{\unicode[STIX]{x1D711},{\mathcal{S}}}(x,y)=(Tx,S_{\unicode[STIX]{x1D711}(x)}(y))$, where $\unicode[STIX]{x1D707}$ stands for the Lebesgue measure on $\mathbb{T}$ and $\unicode[STIX]{x1D708}$ denotes the unique ${\mathcal{S}}$-invariant measure, has the property of asymptotically orthogonal powers. This gives a class of relatively weakly mixing extensions of irrational rotations for which Sarnak’s conjecture on the Möbius disjointness holds for all uniquely ergodic models of $T_{\unicode[STIX]{x1D711},{\mathcal{S}}}$. Moreover, we obtain a class of ‘random’ ergodic sequences $(c_{n})\subset \mathbb{Z}$ such that if $\boldsymbol{\unicode[STIX]{x1D707}}$ denotes the Möbius function, then $$\begin{eqnarray}\lim _{N\rightarrow \infty }\frac{1}{N}\mathop{\sum }_{n\leq N}g(S_{c_{n}}y)\boldsymbol{\unicode[STIX]{x1D707}}(n)=0\end{eqnarray}$$ for all (continuous) uniquely ergodic flows ${\mathcal{S}}$, all $g\in C(Y)$ and $y\in Y$.