<p>Thieullen defined <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1333_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-metric as <Equation ID="Equ18"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1333_Article_Equ18.gif" Format="GIF" Height="29" Rendition="HTML" Resolution="72" Type="Linedraw" Width="237" /> </MediaObject> <EquationSource Format="TEX">$ d^\alpha _n(x,y)=\max _{0\le i\le n-1}e^{i\alpha }d(T^ix,T^iy).$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msubsup> <mi>d</mi> <mi>n</mi> <mi>α</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <munder> <mo movablelimits="true">max</mo> <mrow> <mn>0</mn> <mo>≤</mo> <mi>i</mi> <mo>≤</mo> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </munder> <msup> <mi>e</mi> <mrow> <mi>i</mi> <mi>α</mi> </mrow> </msup> <mi>d</mi> <mrow> <mo stretchy="false">(</mo> <msup> <mi>T</mi> <mi>i</mi> </msup> <mi>x</mi> <mo>,</mo> <msup> <mi>T</mi> <mi>i</mi> </msup> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mrow> </math></EquationSource> </Equation>Expanding upon this foundation, we introduce the notations of Pesin-Pitskel <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1333_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-estimation topological pressure and upper capacity <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1333_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-estimation topological pressure for subsets. We establish Billingsley theorems and variational principles for the Pesin-Pitskel <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1333_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-topological pressure of compact subsets, in terms of the lower Brin-Katok and Katok <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1333_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-estimation pressures. Furthermore, by employing techniques from convex analysis and functional analysis, we derive two variational principles involving Borel probability measures and <i>T</i>-invariant measures.</p>

Variational Principles for $\alpha $-estimation Topological Pressure

期刊论文
发表日期：

2025年7月7日

查看全文

Yunxiang Xie,
Fei Gao,
Menglin Ye

摘要

Thieullen defined $\alpha $ $α$ -metric as $ d^\alpha _n(x,y)=\max _{0\le i\le n-1}e^{i\alpha }d(T^ix,T^iy).$ $d_{n}^{α} (x, y) = max_{0 \leq i \leq n - 1} e^{i α} d (T^{i} x, T^{i} y) .$ Expanding upon this foundation, we introduce the notations of Pesin-Pitskel $\alpha $ $α$ -estimation topological pressure and upper capacity $\alpha $ $α$ -estimation topological pressure for subsets. We establish Billingsley theorems and variational principles for the Pesin-Pitskel $\alpha $ $α$ -topological pressure of compact subsets, in terms of the lower Brin-Katok and Katok $\alpha $ $α$ -estimation pressures. Furthermore, by employing techniques from convex analysis and functional analysis, we derive two variational principles involving Borel probability measures and T-invariant measures.

Variational Principles for \(\alpha \)-estimation Topological Pressure

摘要