<p>We consider <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$C^3$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation> unimodal maps with a non-flat critical point of order <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\ell $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation>, all of whose periodic points are hyperbolic repelling. We prove that for any such map <i>f</i> that is only finitely renormalizable, and for any <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$t > 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> for which the Poincaré series <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {P}(c,t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">P</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is finite, the series <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\sum _{n=0}^{\infty } |Df^n(f(c))|^{-t/\ell }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mo>∑</mo> <mrow> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>∞</mi> </msubsup> <msup> <mrow> <mo stretchy="false">|</mo> <mi>D</mi> <msup> <mi>f</mi> <mi>n</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">|</mo> </mrow> <mrow> <mo>-</mo> <mi>t</mi> <mo stretchy="false">/</mo> <mi>ℓ</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> converges. The proof proceeds by showing that such maps satisfy the backward contracting property.</p>

The Convergence of Poincaré Series of Unimodal Maps at the Critical Point Implies Summability

期刊论文
发表日期：

2026年5月22日

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Huaibin Li

摘要

We consider $C^3$ $C^{3}$ unimodal maps with a non-flat critical point of order $\ell $ $ℓ$ , all of whose periodic points are hyperbolic repelling. We prove that for any such map f that is only finitely renormalizable, and for any $t > 0$ $t > 0$ for which the Poincaré series $\mathcal {P}(c,t)$ $P (c, t)$ is finite, the series $\sum _{n=0}^{\infty } |Df^n(f(c))|^{-t/\ell }$ $\sum_{n = 0}^{\infty} {| D f^{n} (f (c)) |}^{- t / ℓ}$ converges. The proof proceeds by showing that such maps satisfy the backward contracting property.