A measure-theoretic pressure was defined by [L. He, J. Lv and L. Zhou: Definition of measure-theoretic pressure using spanning sets, Acta Math. Sinica (English Series) 20, 709–718 (2004)] based on the Katok entropy formula. For a measure preserving map f, we generalized this definition to define a measure-theoretic pressure by using both <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40306_2025_564_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="38" /> </InlineMediaObject> <EquationSource Format="TEX">$(n,\epsilon )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>ϵ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-spanning and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40306_2025_564_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="38" /> </InlineMediaObject> <EquationSource Format="TEX">$(n,\epsilon )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>ϵ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-separated sets. A variational principle for this pressure is established. Furthermore, we investigate an upper bound for the measure theoretic pressure of a <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40306_2025_564_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$C^{2}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> endomorphism preserving a hyperbolic measure.

Measure Pressure for Measure Preserving Maps and an Upper Bound for the Case of $C^2$ Endomorphisms

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2025年3月10日

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Sanaz Lamei,
Pouya Mehdipour,
Maryam Razi

摘要

A measure-theoretic pressure was defined by [L. He, J. Lv and L. Zhou: Definition of measure-theoretic pressure using spanning sets, Acta Math. Sinica (English Series) 20, 709–718 (2004)] based on the Katok entropy formula. For a measure preserving map f, we generalized this definition to define a measure-theoretic pressure by using both $(n,\epsilon )$ $(n, ϵ)$ -spanning and $(n,\epsilon )$ $(n, ϵ)$ -separated sets. A variational principle for this pressure is established. Furthermore, we investigate an upper bound for the measure theoretic pressure of a $C^{2}$ $C^{2}$ endomorphism preserving a hyperbolic measure.

Measure Pressure for Measure Preserving Maps and an Upper Bound for the Case of \(C^2\) Endomorphisms

摘要