<p>We study the recurrence properties of certain skew products over symmetric interval exchange transformations, including rotations, with cocycles of the form <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$f(x)=-\frac{1}{x^a}+\frac{1}{(1-x)^a}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mo>-</mo> <mfrac> <mn>1</mn> <msup> <mi>x</mi> <mi>a</mi> </msup> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>a</mi> </msup> </mfrac> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$a>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. We prove that typically, such systems are dissipative. However, at the same time they are <i>topologically transitive</i>, i.e. for every two open rectangles <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$A,B\subset [0,1)\times \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>⊂</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>×</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation>, there exists an infinite sequence <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$(q_n)_{n=1}^{\infty }$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>q</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>∞</mi> </msubsup> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$T^{q_n}_f(A)\cap B\ne \emptyset $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>T</mi> <mi>f</mi> <msub> <mi>q</mi> <mi>n</mi> </msub> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> <mo>∩</mo> <mi>B</mi> <mo>≠</mo> <mi mathvariant="normal">∅</mi> </mrow> </math></EquationSource> </InlineEquation>.</p>

Coexistence of two contrasting recurrence properties of certain non-integrable cocycles

期刊论文
出版模式：

开放获取
发表日期：

2026年7月27日

查看全文

Przemysław Berk,
Ł.ukasz Kotlewski

摘要

We study the recurrence properties of certain skew products over symmetric interval exchange transformations, including rotations, with cocycles of the form $f(x)=-\frac{1}{x^a}+\frac{1}{(1-x)^a}$ $f (x) = - \frac{1}{x^{a}} + \frac{1}{{(1 - x)}^{a}}$ , where $a>1$ $a > 1$ . We prove that typically, such systems are dissipative. However, at the same time they are topologically transitive, i.e. for every two open rectangles $A,B\subset [0,1)\times \mathbb {R}$ $A, B \subset [0, 1) \times R$ , there exists an infinite sequence $(q_n)_{n=1}^{\infty }$ ${(q_{n})}_{n = 1}^{\infty}$ such that $T^{q_n}_f(A)\cap B\ne \emptyset $ $T_{f}^{q_{n}} (A) \cap B \neq \emptyset$ .