Rigidity in dynamics and Mbius disjointness Mariusz Lemaczyk - PowerPoint PPT Presentation

Rigidity in dynamics and Möbius disjointness Mariusz Lemańczyk Nicolaus Copernicus University, Toruń Nombre premiers, déterminisme et pseudoaléa, CIRM Marseille, 4-8.11.2019 based on a joint work with Adam Kanigowski and Maksym Radziwiłł 1 / 24

Sarnak’s conjecture Definition (Möbius disjointness) Let X be a compact metric space and T : X → X a homeomorphism of it. One says that ( X , T ) is Möbius disjoint if � lim N →∞ 1 n � N f ( T n x ) µ ( n ) = 0 for all x ∈ X , f ∈ C ( X ) . N Sarnak’s conjecture (2010) All zero topological entropy dynamical systems ( X , T ) are Möbius disjoint. Remark: Möbius disjointness of ( X , T ) is often rephrased as Sarnak’s conjecture is satisfied for a topological system ( X , T ) or that the homeomorphism T is Möbius orthogonal . 2 / 24

Sarnak’s conjecture Definition (Möbius disjointness) Let X be a compact metric space and T : X → X a homeomorphism of it. One says that ( X , T ) is Möbius disjoint if � lim N →∞ 1 n � N f ( T n x ) µ ( n ) = 0 for all x ∈ X , f ∈ C ( X ) . N Sarnak’s conjecture (2010) All zero topological entropy dynamical systems ( X , T ) are Möbius disjoint. Remark: Möbius disjointness of ( X , T ) is often rephrased as Sarnak’s conjecture is satisfied for a topological system ( X , T ) or that the homeomorphism T is Möbius orthogonal . 2 / 24

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How do people show Möbius disjointness in selected classes? � µ is aperiodic (classical), that is: 1 n � N µ ( an + b ) → 0 for N each a , b � 1. Check that each periodic system is Möbius disjoint; this can be used in a weaker version when a system is “approximated” by periodic systems: Karagulyan for zero entropy interval maps, 2013; subshifts given by regular Toeplitz sequences: El Abdalaoui, Downarowicz, Kasjan, L., 2013, synchronized automata: Deshouilliers, Drmota, Müllner, 2015 . Daboussi-Delange-Kátai-Bourgain-Sarnak-Ziegler criterion (DDKBSZ): ( f n ) ⊂ C bounded such that � 1 n � N f p 1 n f p 2 n → 0 for each different primes p 1 , p 2 sufficiently large then � N n � N f n u ( n ) → 0 for EACH bounded multiplicative u : N → C . In the dynamical context ( X , T ) we use it for f n = f ( T n x ) (Bourgain, Sarnak, Ziegler - Möbius disjointness of horocycle flows, 2013). Applied when T p 1 and T p 2 are disjoint (or approximately disjoint) in the Furstenberg sense. Matomäki and Radziwiłł, 2015: For the Möbius function we have cancellations on typical short intervals: � � � � � � 1 � 1 h � H µ ( m + h ) � → 0 when H , M → ∞ and H = o ( M ) . M m � M H 7 / 24

How do people show Möbius disjointness in selected classes? � µ is aperiodic (classical), that is: 1 n � N µ ( an + b ) → 0 for N each a , b � 1. Check that each periodic system is Möbius disjoint; this can be used in a weaker version when a system is “approximated” by periodic systems: Karagulyan for zero entropy interval maps, 2013; subshifts given by regular Toeplitz sequences: El Abdalaoui, Downarowicz, Kasjan, L., 2013, synchronized automata: Deshouilliers, Drmota, Müllner, 2015 . Daboussi-Delange-Kátai-Bourgain-Sarnak-Ziegler criterion (DDKBSZ): ( f n ) ⊂ C bounded such that � 1 n � N f p 1 n f p 2 n → 0 for each different primes p 1 , p 2 sufficiently large then � N n � N f n u ( n ) → 0 for EACH bounded multiplicative u : N → C . In the dynamical context ( X , T ) we use it for f n = f ( T n x ) (Bourgain, Sarnak, Ziegler - Möbius disjointness of horocycle flows, 2013). Applied when T p 1 and T p 2 are disjoint (or approximately disjoint) in the Furstenberg sense. Matomäki and Radziwiłł, 2015: For the Möbius function we have cancellations on typical short intervals: � � � � � � 1 � 1 h � H µ ( m + h ) � → 0 when H , M → ∞ and H = o ( M ) . M m � M H 7 / 24

How do people show Möbius disjointness in selected classes? � µ is aperiodic (classical), that is: 1 n � N µ ( an + b ) → 0 for N each a , b � 1. Check that each periodic system is Möbius disjoint; this can be used in a weaker version when a system is “approximated” by periodic systems: Karagulyan for zero entropy interval maps, 2013; subshifts given by regular Toeplitz sequences: El Abdalaoui, Downarowicz, Kasjan, L., 2013, synchronized automata: Deshouilliers, Drmota, Müllner, 2015 . Daboussi-Delange-Kátai-Bourgain-Sarnak-Ziegler criterion (DDKBSZ): ( f n ) ⊂ C bounded such that � 1 n � N f p 1 n f p 2 n → 0 for each different primes p 1 , p 2 sufficiently large then � N n � N f n u ( n ) → 0 for EACH bounded multiplicative u : N → C . In the dynamical context ( X , T ) we use it for f n = f ( T n x ) (Bourgain, Sarnak, Ziegler - Möbius disjointness of horocycle flows, 2013). Applied when T p 1 and T p 2 are disjoint (or approximately disjoint) in the Furstenberg sense. Matomäki and Radziwiłł, 2015: For the Möbius function we have cancellations on typical short intervals: � � � � � � 1 � 1 h � H µ ( m + h ) � → 0 when H , M → ∞ and H = o ( M ) . M m � M H 7 / 24

How do people show Möbius disjointness in selected classes? � µ is aperiodic (classical), that is: 1 n � N µ ( an + b ) → 0 for N each a , b � 1. Check that each periodic system is Möbius disjoint; this can be used in a weaker version when a system is “approximated” by periodic systems: Karagulyan for zero entropy interval maps, 2013; subshifts given by regular Toeplitz sequences: El Abdalaoui, Downarowicz, Kasjan, L., 2013, synchronized automata: Deshouilliers, Drmota, Müllner, 2015 . Daboussi-Delange-Kátai-Bourgain-Sarnak-Ziegler criterion (DDKBSZ): ( f n ) ⊂ C bounded such that � 1 n � N f p 1 n f p 2 n → 0 for each different primes p 1 , p 2 sufficiently large then � N n � N f n u ( n ) → 0 for EACH bounded multiplicative u : N → C . In the dynamical context ( X , T ) we use it for f n = f ( T n x ) (Bourgain, Sarnak, Ziegler - Möbius disjointness of horocycle flows, 2013). Applied when T p 1 and T p 2 are disjoint (or approximately disjoint) in the Furstenberg sense. Matomäki and Radziwiłł, 2015: For the Möbius function we have cancellations on typical short intervals: � � � � � � 1 � 1 h � H µ ( m + h ) � → 0 when H , M → ∞ and H = o ( M ) . M m � M H 7 / 24

How do people show Möbius disjointness in selected classes? � µ is aperiodic (classical), that is: 1 n � N µ ( an + b ) → 0 for N each a , b � 1. Check that each periodic system is Möbius disjoint; this can be used in a weaker version when a system is “approximated” by periodic systems: Karagulyan for zero entropy interval maps, 2013; subshifts given by regular Toeplitz sequences: El Abdalaoui, Downarowicz, Kasjan, L., 2013, synchronized automata: Deshouilliers, Drmota, Müllner, 2015 . Daboussi-Delange-Kátai-Bourgain-Sarnak-Ziegler criterion (DDKBSZ): ( f n ) ⊂ C bounded such that � 1 n � N f p 1 n f p 2 n → 0 for each different primes p 1 , p 2 sufficiently large then � N n � N f n u ( n ) → 0 for EACH bounded multiplicative u : N → C . In the dynamical context ( X , T ) we use it for f n = f ( T n x ) (Bourgain, Sarnak, Ziegler - Möbius disjointness of horocycle flows, 2013). Applied when T p 1 and T p 2 are disjoint (or approximately disjoint) in the Furstenberg sense. Matomäki and Radziwiłł, 2015: For the Möbius function we have cancellations on typical short intervals: � � � � � � 1 � 1 h � H µ ( m + h ) � → 0 when H , M → ∞ and H = o ( M ) . M m � M H 7 / 24

Möbius disjointness - general results Theorem (Huang, Wang, Zhang, 2016) Möbius disjointness holds for each ( X , T ) for which EACH invariant measure yields a measure-theoretic system with discrete spectrum. Short interval behaviour of µ is used. In fact, as shown later by Ferenczi, Kułaga-Przymus and L., an interpretation of a result on averaged form of Chowla conjecture 1 (for the Möbius function) by Matomäki, Radziwiłł and Tao from 2015 gives that the spectral measure of the projection on the zero-coordinate in each Furstenberg system must be continuous. Discrete spectrum result immediately follows. 1 � � � � � � � � λ ( n ) λ ( n + h ) � = o ( HX ) � � � h � H n � X provided that H → ∞ arbitrarily slowly with X → ∞ (and H � X ) 8 / 24

Möbius disjointness - general results Theorem (Huang, Wang, Zhang, 2016) Möbius disjointness holds for each ( X , T ) for which EACH invariant measure yields a measure-theoretic system with discrete spectrum. Short interval behaviour of µ is used. In fact, as shown later by Ferenczi, Kułaga-Przymus and L., an interpretation of a result on averaged form of Chowla conjecture 1 (for the Möbius function) by Matomäki, Radziwiłł and Tao from 2015 gives that the spectral measure of the projection on the zero-coordinate in each Furstenberg system must be continuous. Discrete spectrum result immediately follows. 1 � � � � � � � � λ ( n ) λ ( n + h ) � = o ( HX ) � � � h � H n � X provided that H → ∞ arbitrarily slowly with X → ∞ (and H � X ) 8 / 24