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• W1989181805 abstract "Let<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f><mml:semantics><mml:mi>f</mml:mi><mml:annotation encoding=application/x-tex>f</mml:annotation></mml:semantics></mml:math></inline-formula>be a<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C Superscript 1 plus epsilon><mml:semantics><mml:msup><mml:mi>C</mml:mi><mml:mrow class=MJX-TeXAtom-ORD><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>ε<!-- ε --></mml:mi></mml:mrow></mml:msup><mml:annotation encoding=application/x-tex>C^{1+varepsilon }</mml:annotation></mml:semantics></mml:math></inline-formula>diffeomorphism on a compact smooth surface with positive topological entropy<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=h><mml:semantics><mml:mi>h</mml:mi><mml:annotation encoding=application/x-tex>h</mml:annotation></mml:semantics></mml:math></inline-formula>. For every<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=0 greater-than delta greater-than h><mml:semantics><mml:mrow><mml:mn>0</mml:mn><mml:mo>></mml:mo><mml:mi>δ<!-- δ --></mml:mi><mml:mo>></mml:mo><mml:mi>h</mml:mi></mml:mrow><mml:annotation encoding=application/x-tex>0>delta >h</mml:annotation></mml:semantics></mml:math></inline-formula>, we construct an invariant Borel set<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper E><mml:semantics><mml:mi>E</mml:mi><mml:annotation encoding=application/x-tex>E</mml:annotation></mml:semantics></mml:math></inline-formula>and a countable Markov partition for the restriction of<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f><mml:semantics><mml:mi>f</mml:mi><mml:annotation encoding=application/x-tex>f</mml:annotation></mml:semantics></mml:math></inline-formula>to<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper E><mml:semantics><mml:mi>E</mml:mi><mml:annotation encoding=application/x-tex>E</mml:annotation></mml:semantics></mml:math></inline-formula>in such a way that<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper E><mml:semantics><mml:mi>E</mml:mi><mml:annotation encoding=application/x-tex>E</mml:annotation></mml:semantics></mml:math></inline-formula>has full measure with respect to every ergodic invariant probability measure with entropy greater than<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=delta><mml:semantics><mml:mi>δ<!-- δ --></mml:mi><mml:annotation encoding=application/x-tex>delta</mml:annotation></mml:semantics></mml:math></inline-formula>. The following results follow:<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f><mml:semantics><mml:mi>f</mml:mi><mml:annotation encoding=application/x-tex>f</mml:annotation></mml:semantics></mml:math></inline-formula>has at most countably many ergodic measures of maximal entropy (a conjecture of J. Buzzi), and in the case when<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f><mml:semantics><mml:mi>f</mml:mi><mml:annotation encoding=application/x-tex>f</mml:annotation></mml:semantics></mml:math></inline-formula>is<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C Superscript normal infinity><mml:semantics><mml:msup><mml:mi>C</mml:mi><mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi></mml:msup><mml:annotation encoding=application/x-tex>C^infty</mml:annotation></mml:semantics></mml:math></inline-formula>,<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=limit sup e Superscript minus n h Baseline number-sign StartSet x colon f Superscript n Baseline left-parenthesis x right-parenthesis equals x EndSet greater-than 0><mml:semantics><mml:mrow><mml:munder><mml:mo form=prefix>lim sup</mml:mo><mml:mrow class=MJX-TeXAtom-ORD><mml:mi>n</mml:mi><mml:mo stretchy=false>→<!-- → --></mml:mo><mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi></mml:mrow></mml:munder><mml:msup><mml:mi>e</mml:mi><mml:mrow class=MJX-TeXAtom-ORD><mml:mo>−<!-- − --></mml:mo><mml:mi>n</mml:mi><mml:mi>h</mml:mi></mml:mrow></mml:msup><mml:mi mathvariant=normal>#<!-- # --></mml:mi><mml:mo fence=false stretchy=false>{</mml:mo><mml:mi>x</mml:mi><mml:mo>:</mml:mo><mml:msup><mml:mi>f</mml:mi><mml:mi>n</mml:mi></mml:msup><mml:mo stretchy=false>(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy=false>)</mml:mo><mml:mo>=</mml:mo><mml:mi>x</mml:mi><mml:mo fence=false stretchy=false>}</mml:mo><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:annotation encoding=application/x-tex>limsup limits _{nto infty }e^{-n h}#{x:f^n(x)=x}>0</mml:annotation></mml:semantics></mml:math></inline-formula>(a conjecture of A. Katok)." @default.
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• W1989181805 title "Symbolic dynamics for surface diffeomorphisms with positive entropy" @default.
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