Let $\lambda$ denote the probability Lebesgue measure on ${\mathbb T}^2$. For
any $C^2$-Anosov diffeomorphism of the $2$-torus preserving $\lambda$ with
measure-theoretic entropy equal to topological entropy, we show that the set of
points with nondense orbits is hyperplane absolute winning (HAW). This
generalizes the result in~\cite[Theorem~1.4]{T4} for $C^2$-expanding maps of
the circle.Comment: Minor typos corrected. Added more expositio

Tseng, Jimmy

English

arXiv

Let λ denote the probability Lebesgue measure on Τ2. For any C2-Anosov diffeomorphism of the 2-torus preserving λ with measure-theoretic entropy equal to topological entropy,we show that the set of points with nondense orbits is hyperplane absolute winning (HAW). This generalizes the result in [18, Theorem 1.4] for C2-expanding maps of the circle

Explore Bristol Research

                          Tseng, J. (2016). Nondense orbits for Anosov diffeomorphisms of the 2-torus. Real Analysis Exchange, 41(2), 307-314.Peer reviewed versionLink to publication record in Explore Bristol ResearchPDF-documentThis is the author accepted manuscript (AAM). The final published version (version of record) is available onlinevia Michigan State University Press at http://msupress.org/journals/issue/?id=50-21D-608 . Please refer to anyapplicable terms of use of the publisher.University of Bristol - Explore Bristol ResearchGeneral rightsThis document is made available in accordance with publisher policies. Please cite only the publishedversion using the reference above. Full terms of use are available:http://www.bristol.ac.uk/pure/about/ebr-terms.htmlNONDENSE ORBITS FOR ANOSOV DIFFEOMORPHISMS OF THE 2-TORUSJIMMY TSENGAbstract. Let λ denote the probability Lebesgue measure on T2. For any C2-Anosov diffeo-morphism of the 2-torus preserving λ with measure-theoretic entropy equal to topological entropy,we show that the set of points with nondense orbits is hyperplane absolute winning (HAW). Thisgeneralizes the result in [18, Theorem 1.4] for C2-expanding maps of the circle.1. IntroductionFor a dynamical system f : X → X on a set X with a topology, we say that a point has nondenseforward orbit if its forward orbit closure is a proper subset of X, and we call the set of these pointsthe nondense set and denote it by ND(f). The study of nondense orbits is important for theinteractions between ergodic theory and Diophantine approximation and, partly inspired by this,there has been much recent interest in these sets (see [1, 3, 5, 6, 15, 18, 20, 21] for example). Ofparticular interest is the size of ND(f) and whether these sets have the winning property, which isa strengthening of having full Hausdorff dimension and is stable under intersections. (See Section 2for the precise definition of the winning property.) Almost all of the known dynamical systemsthat have winning nondense sets are (piecewise) linear or come from automorphisms. A class ofexceptions is the result by the author for C2-expanding self-maps on the circle [18, Theorem 1.4].There is also a recent result by W. Wu [21, Section 2] for partially hyperbolic diffeomorphismson compact manifolds in which the diffeomorphisms are conformal when restricted to the unstablemanifolds or the unstable manifolds are one-dimensional. Wu’s result, however, shows only thatthe nondense set when restricted to an unstable manifold is winning, but does not show that thewhole nondense set is winning.1 General diffeomorphisms present a challenge in dimensions higherthan one for determining whether the nondense set is winning or not.1.1. Statement of results. The purpose of this brief note is to show that the members of a large,natural class of nonlinear self-maps on the 2-torus have winning nondense sets, thereby, givinga large, natural class of higher dimensional examples of winning nondense sets for more generaldiffeomorphisms. Since the winning property is stable under intersections, our results, in fact, alsoapply to nondense complete orbits. Let λ denote the probability Lebesgue measure on T2. Ourmain result is the following.Theorem 1.1. Let g : T2 → T2 be a C2-Anosov diffeomorphism preserving λ such that hλ(g) =htop(g). Let x0 ∈ T2. Then the setND(g,x0) :={x ∈ T2 : x0 /∈ {gn(x)}∞n=0}is hyperplane absolute winning (HAW) and thus has full Hausdorff dimension.The author acknowledges the research leading to these results has received funding from the European ResearchCouncil under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n.291147.1Wu’s result does imply that the whole nondense set has full Hausdorff dimension, which is a significant result. Weare, however, interested in the strengthening to winning because of its stability under intersections. In particular, ourresult, Theorem 1.1, applies to the smaller set of points with nondense complete orbits because the winning propertyis stable under intersections and shows that this smaller set also has full Hausdorff dimension.12 JIMMY TSENGNote that the hyperplane absolute winning property is a variant of the winning property and impliesit (see Section 2).Corollary 1.2. Let {gm}∞m=0 be a family of C2-Anosov diffeomorphisms preserving λ such thathλ(gm) = htop(gm) for all m. Let {x`}∞`=0 ⊂ T2. Then the set∞⋂m=0∞⋂`=0(ND(gm,x`) ∩ND(g−1m ,x`))(1.1)is hyperplane absolute winning (HAW) and thus has full Hausdorff dimension.Remark 1.3. Since the countable intersection property holds for all HAW subsets on T2 (seeSection 2), we may intersect the set from (1.1) with any countable family of HAW subsets onT2 and still retain the HAW property and thus also retain the property of having full Hausdorffdimension. For example, the intersection of the set from (1.1) with any countable collection oftranslates of badly approximable vectors on T2 (which is HAW [4, Theorem 2.5]) is still HAW and,thus, has full Hausdorff dimension.Remark 1.4. We can also obtain results when the dimension d is greater than two. It followsfrom the proof of Theorem 1.1 and from (4) of Lemma 2.1 that any Anosov diffeomorphism havinga bilipschitz (or quasisymmetric) conjugacy with a hyperbolic automorphism on Td will have anα-winning nondense set for some α > 0 (see also Section 3.1).Theorem 1.1 answers the question, first raised in [18, Section 6], of whether there are nonlineardynamical systems with winning nondense sets in dimensions greater than one.2. Winning and hyperplane absolute winningThe winning property was introduced by W. Schmidt [17] in 1966 and has many later variants(see [4, 10, 16]). We define the winning property and one particular strengthening, the hyperplaneabsolute winning (HAW) property, for Rd. HAW was introduced in [4].Let 0 < α < 1 and 0 < β < 1. Let S ⊂ Rd and ρ(·) denote the radius of a closed ball. Twoplayers, Alice A and Bob B, alternate choosing nested closed ballsB1 ⊃ A1 ⊃ B2 ⊃ A2 · · ·on Rd according to the following rules:ρ(An) = αρ(Bn) and ρ(Bn) = βρ(An−1). (2.1)A set S is called (α, β)-winning if Alice has a strategy guaranteeing ∩iBi lies in S for the givenα and β. A set S is called α-winning if it is (α, β)-winning for the given α and every β. A setS is called winning if it is α-winning for some α. These sets have four important properties forus [16, 17]:Lemma 2.1. Properties of winning sets.(1) An α-winning set in Rn is dense and of full Hausdorff dimension.(2) A countable intersection of α-winning sets is α-winning.(3) An α-winning set in Rn with a countable number of points removed is α-winning.(4) The image of an α-winning set under a bilipschitz map is α′-winning, where α′ dependsonly on α and the map.Now fix a k ∈ {0, 1, · · · , d−1} and restrict 0 < β < 1/3. Bob initially chooses x1 ∈ Rd and r1 > 0and forms a closed ball B1 = B(x1, r1). At each stage of the game, after Bob chooses xi ∈ Rdand ri > 0, Alice chooses an affine subspace L of dimension k and removes an εi-neighborhoodAi = L(εi) from Bi := B(xi, ri) for some 0 < εi ≤ βri. Then Bob chooses xi+1 and ri+1 ≥ βri suchthatBi+1 := B(xi+1, ri+1) ⊂ Bi\Ai.NONDENSE ORBITS FOR ANOSOV DIFFEOMORPHISMS OF THE 2-TORUS 3A set S is said to be k-dimensionally β-absolute winning if Alice has a strategy guaranteeing that∩iBi intersects S. A set S is k-dimensionally absolute winning if it is k-dimensionally β-absolutewinning for every 0 < β < 1/3. We call (d − 1)-dimensionally absolute winning sets hyperplaneabsolute winning (HAW) sets. These sets have three important properties for us [4, Proposition 2.3]:Lemma 2.2. Properties of k-dimensionally absolute winning sets.(1) HAW (and thus k-dimensional absolute winning for all 0 ≤ k ≤ d − 1) implies α-winningfor all 0 < α < 1/2.(2) The countable intersection of k-dimensionally absolute winning sets is k-dimensionally ab-solute winning.(3) The image of a k-dimensionally absolute winning set under a C1 diffeomorphism of Rd isk-dimensionally absolute winning.3. ProofsIn this section, we prove Theorem 1.1 and its corollary. The two main ingredients are the HAWproperty, already introduced, and the following smooth classification of Anosov diffeomorphisms ofthe 2-torus (see [9, Corollary 20.4.5] for example):Theorem 3.1. Suppose g : T2 → T2 is a C2-Anosov diffeomorphism preserving λ such that hλ(g) =htop(g). Then g is C1 conjugate to a linear automorphism.Proof of Theorem 1.1. The Anosov diffeomorphism g is C1-conjugate to a linear automorphism Tby Theorem 3.1. Let h : T2 → T2 be this conjugacy from the Anosov diffeomorphism to thelinear automorphism. The nondense set ND(T, h(x0)) is HAW by [4, Theorem 2.6]. Consequently,ND(g,x0) is HAW by (3) from Lemma 2.2. This proves the desired result. Proof of Corollary 1.2. The desired result follows from the theorem and (1) and (2) from Lemma 2.2and (1) from Lemma 2.1. 3.1. Remarks. A key ingredient in the proof of the smooth classification is the fact that the stableand unstable foliations for a C2-Anosov diffeomorphism of the 2-torus are C1. The proof of thefact relies on a more general result, namely that a codimension-1 stable manifold for a C2-Anosovdiffeomorphism of a compact manifold forms a C1 foliation [8, Corollary 4]. The same applies tothe unstable manifold, from which we obtain the desired C1 foliations for the 2-torus. This leadsto a characterization of smooth conjugacies for the 2-torus [11, 13, 14] (also see [9, Chapter 20.4.b])and to our smooth classification. (Note that, for a torus of dimension three or greater, either theunstable manifold or the stable manifold will not be codimension-1.)For dimensions three and greater, we lack a smooth classification theorem. In fact, the charac-terization of smooth conjugacies is more complicated, and R. de la Llave showed that two Anosovdiffeomorphisms of Td for d ≥ 4 may be topologically conjugate without being Lipschitz conju-gate [12, Theorem 6.3]. (For the 3-torus, the situation is not settled.) A. Gogolev noted that de laLlave’s counterexample can be generalized so that (in most cases) one can take one of the diffeo-morphisms to be a linear automorphism [7, Theorem B]. Should a smooth classification in higherdimensions be found, it would be more complicated than Theorem 3.1, especially since it mustallow for cases in which the conjugacy is only Ho¨lder. On the other hand, we always have a Ho¨lderclassification theorem for this context: every Anosov diffeomorphism of Td is Ho¨lder conjugate toa linear hyperbolic automorphism of Td [9, Theorems 18.6.1 and 19.1.2]. This Ho¨lder classificationis, however, not strong enough for us because we need the conjugacy to be quasisymmetric orbilipschitz (see [16] and (4) from Lemma 2.1) to carry out the proof in this paper.Another way of approaching this problem for dimensions three and greater that should avoidusing a smooth classification is to employ a technique used earlier [18, 21], namely using a boundeddistortion property (i.e. universal bounds on the ratio of the absolute value of the Jacobian deter-minant for nearby points—see [18, Section 2.1.1] for example). If we assume conformality, bounded4 JIMMY TSENGdistortion follows. However, without this assumption (in dimensions two or greater), bounded dis-tortion may not follow, and without it we may not be able to control the local “shape” of successiveiterates of small balls. Since the variants of the game involve intersections with balls, this lack ofcontrol is an obstacle that needs to be overcome if this approach were to succeed for more generalnonlinear maps.Also, we note that for linear maps, one can replace bounded distortion with linear properties ofthe map. In particular, one can decompose Rd into a direct sum of subspaces appropriate for thelinear action and, using the HAW property, one is allowed to remove “bad” affine hyperplanes [4].Linearity seems important to this step and may not possible with a nonlinear system.Finally, we mention a theorem closely related to our results. For pairs of toral mappings, onecan consider the set of points with orbits that are nondense under one of the pair and dense underthe other, a problem first considered in [2]. The nonlinear case for this setup of simultaneous denseand nondense orbits is also interesting, and, for generic pairs of C2-Anosov diffeomorphisms of T2,it is known that the desired set is uncountable and dense [19, Theorem 1.3].References[1] A. G. Abercrombie and R. Nair, An exceptional set in the ergodic theory of expanding maps on manifolds,Monatsh. Math. 148 (2006), 1–17.[2] V. Bergelson, M. Einsiedler, and J. Tseng, Simultaneous dense and nondense orbits for commuting maps, IsraelJ. Math. 210 (2015), 23–45.[3] R. Broderick, L. Fishman, and D. Kleinbock, Schmidt’s game, fractals, and orbits of toral endomorphisms,Ergodic Theory Dynam. Systems 31 (2011), 1095–1107.[4] R. Broderick, L. Fishman, D. Kleinbock, A. Reich, and B. Weiss, The set of badly approximable vectors is stronglyC1 incompressible. Math. Proc. Cambridge Philos. Soc. 153 (2012), no. 2, 319–339.[5] S. G. Dani, On orbits of endomorphisms of tori and the Schmidt game, Ergodic Theory Dynam. Systems 8(1988), 523-529.[6] D. Fa¨rm, Simultaneously non-dense orbits under different expanding maps. Dyn. Syst. 25 (2010), 531–545.[7] A. Gogolev, Smooth conjugacy of Anosov diffeomorphisms on higher-dimensional tori, J. Mod. Dyn. 2 (2008),no. 4, 645-700.[8] M. W. Hirsch and C. C. Pugh, Stable manifolds and hyperbolic sets. 1970 Global Analysis (Proc. Sympos. PureMath., Vol. XIV, Berkeley, Calif., 1968) pp. 133-163 Amer. Math. Soc., Providence, R.I.[9] A. Katok and B. Hasselblatt, “Introduction to the modern theory of dynamical systems. With a supplementarychapter by Katok and Leonardo Mendoza.” Encyclopedia of Mathematics and its Applications, 54. CambridgeUniversity Press, Cambridge, 1995.[10] D. Kleinbock and B. Weiss, Modified Schmidt games and Diophantine approximation with weights, Adv. Math.223 (2010), 1276–1298.[11] R. de la Llave, Invariants for smooth conjugacy of hyperbolic dynamical systems. II. Comm. Math. Phys. 109(1987), no. 3, 369–378.[12] R. de la Llave, Smooth conjugacy and S-R-B measures for uniformly and non-uniformly hyperbolic systems,Comm. Math. Phys. 150 (1992), no. 2, 289–320.[13] J. M. Marco and R. Moriyn, Invariants for smooth conjugacy of hyperbolic dynamical systems. I. Comm. Math.Phys. 109 (1987), no. 4, 681–689.[14] J. M. Marco and R. Moriyn, Invariants for smooth conjugacy of hyperbolic dynamical systems. III. Comm. Math.Phys. 112 (1987), no. 2, 317–333.[15] B. Mance and J. Tseng, Bounded Lu¨roth expansions: applying Schmidt games where infinite distortion exists,Acta Arith. 158 (2013), 33–47.[16] C. McMullen, Winning sets, quasiconformal maps and Diophantine approximation, Geom. Funct. Anal. 20 2010,726–740.[17] W. Schmidt, Badly approximable numbers and certain games, Trans. Amer. Math. Soc. 123 (1966), 178–199.[18] J. Tseng, Schmidt games and Markov partitions, Nonlinearity 22 (2009), 525-543.[19] J. Tseng, Simultaneous dense and nondense orbits for toral diffeomorphisms, Ergodic Theory and DynamicalSystems, to appear. http://arxiv.org/abs/1406.1970[20] M. Urban´ski, The Hausdorff dimension of the set of points with nondense orbit under a hyperbolic dynamicalsystem, Nonlinearity 4 (1991), 385–397.[21] W. Wu, Schmidt games and non-dense forward orbits of certain partially hyperbolic systems, Ergodic Theoryand Dynamical Systems, to appear. http://dx.doi.org/10.1017/etds.2014.136NONDENSE ORBITS FOR ANOSOV DIFFEOMORPHISMS OF THE 2-TORUS 5School of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW UKE-mail address: j.tseng@bristol.ac.uk jimmytseng01@gmail.com

Nondense orbits for Anosov diffeomorphisms of the 2-torus

null Jimmy Tseng

Crossref

Real Analysis Exchange

Nondense Orbits for Anosov Diffeomorphisms of the 2-Torus

This is the author accepted manuscript. The final version is available from Michigan State University Press via the DOI in this recordLet λ denote the probability Lebesgue measure on T2
. For any C2-Anosov diffeomorphism of the 2-torus preserving λ with measure-theoretic entropy equal to topological entropy,
we show that the set of points with nondense orbits is hyperplane absolute winning (HAW). This generalizes the result in [18, Theorem 1.4] for C2-expanding maps of the circle.The author acknowledges the research leading to these results has received funding from the European Research
Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n.
291147

Jimmy Tseng

Open Research Exeter

NONDENSE ORBITS FOR ANOSOV DIFFEOMORPHISMS OF THE 2-TORUSJIMMY TSENGAbstract. Let λ denote the probability Lebesgue measure on T2. For any C2-Anosov diffeo-morphism of the 2-torus preserving λ with measure-theoretic entropy equal to topological entropy,we show that the set of points with nondense orbits is hyperplane absolute winning (HAW). Thisgeneralizes the result in [18, Theorem 1.4] for C2-expanding maps of the circle.1. IntroductionFor a dynamical system f : X → X on a set X with a topology, we say that a point has nondenseforward orbit if its forward orbit closure is a proper subset of X, and we call the set of these pointsthe nondense set and denote it by ND(f). The study of nondense orbits is important for theinteractions between ergodic theory and Diophantine approximation and, partly inspired by this,there has been much recent interest in these sets (see [1, 3, 5, 6, 15, 18, 20, 21] for example). Ofparticular interest is the size of ND(f) and whether these sets have the winning property, which isa strengthening of having full Hausdorff dimension and is stable under intersections. (See Section 2for the precise definition of the winning property.) Almost all of the known dynamical systemsthat have winning nondense sets are (piecewise) linear or come from automorphisms. A class ofexceptions is the result by the author for C2-expanding self-maps on the circle [18, Theorem 1.4].There is also a recent result by W. Wu [21, Section 2] for partially hyperbolic diffeomorphismson compact manifolds in which the diffeomorphisms are conformal when restricted to the unstablemanifolds or the unstable manifolds are one-dimensional. Wu’s result, however, shows only thatthe nondense set when restricted to an unstable manifold is winning, but does not show that thewhole nondense set is winning.1 General diffeomorphisms present a challenge in dimensions higherthan one for determining whether the nondense set is winning or not.1.1. Statement of results. The purpose of this brief note is to show that the members of a large,natural class of nonlinear self-maps on the 2-torus have winning nondense sets, thereby, givinga large, natural class of higher dimensional examples of winning nondense sets for more generaldiffeomorphisms. Since the winning property is stable under intersections, our results, in fact, alsoapply to nondense complete orbits. Let λ denote the probability Lebesgue measure on T2. Ourmain result is the following.Theorem 1.1. Let g : T2 → T2 be a C2-Anosov diffeomorphism preserving λ such that hλ(g) =htop(g). Let x0 ∈ T2. Then the setND(g,x0) :={x ∈ T2 : x0 /∈ {gn(x)}∞n=0}is hyperplane absolute winning (HAW) and thus has full Hausdorff dimension.The author acknowledges the research leading to these results has received funding from the European ResearchCouncil under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n.291147.1Wu’s result does imply that the whole nondense set has full Hausdorff dimension, which is a significant result. Weare, however, interested in the strengthening to winning because of its stability under intersections. In particular, ourresult, Theorem 1.1, applies to the smaller set of points with nondense complete orbits because the winning propertyis stable under intersections and shows that this smaller set also has full Hausdorff dimension.12 JIMMY TSENGNote that the hyperplane absolute winning property is a variant of the winning property and impliesit (see Section 2).Corollary 1.2. Let {gm}∞m=0 be a family of C2-Anosov diffeomorphisms preserving λ such thathλ(gm) = htop(gm) for all m. Let {x`}∞`=0 ⊂ T2. Then the set∞⋂m=0∞⋂`=0(ND(gm,x`) ∩ND(g−1m ,x`))(1.1)is hyperplane absolute winning (HAW) and thus has full Hausdorff dimension.Remark 1.3. Since the countable intersection property holds for all HAW subsets on T2 (seeSection 2), we may intersect the set from (1.1) with any countable family of HAW subsets onT2 and still retain the HAW property and thus also retain the property of having full Hausdorffdimension. For example, the intersection of the set from (1.1) with any countable collection oftranslates of badly approximable vectors on T2 (which is HAW [4, Theorem 2.5]) is still HAW and,thus, has full Hausdorff dimension.Remark 1.4. We can also obtain results when the dimension d is greater than two. It followsfrom the proof of Theorem 1.1 and from (4) of Lemma 2.1 that any Anosov diffeomorphism havinga bilipschitz (or quasisymmetric) conjugacy with a hyperbolic automorphism on Td will have anα-winning nondense set for some α > 0 (see also Section 3.1).Theorem 1.1 answers the question, first raised in [18, Section 6], of whether there are nonlineardynamical systems with winning nondense sets in dimensions greater than one.2. Winning and hyperplane absolute winningThe winning property was introduced by W. Schmidt [17] in 1966 and has many later variants(see [4, 10, 16]). We define the winning property and one particular strengthening, the hyperplaneabsolute winning (HAW) property, for Rd. HAW was introduced in [4].Let 0 < α < 1 and 0 < β < 1. Let S ⊂ Rd and ρ(·) denote the radius of a closed ball. Twoplayers, Alice A and Bob B, alternate choosing nested closed ballsB1 ⊃ A1 ⊃ B2 ⊃ A2 · · ·on Rd according to the following rules:ρ(An) = αρ(Bn) and ρ(Bn) = βρ(An−1). (2.1)A set S is called (α, β)-winning if Alice has a strategy guaranteeing ∩iBi lies in S for the givenα and β. A set S is called α-winning if it is (α, β)-winning for the given α and every β. A setS is called winning if it is α-winning for some α. These sets have four important properties forus [16, 17]:Lemma 2.1. Properties of winning sets.(1) An α-winning set in Rn is dense and of full Hausdorff dimension.(2) A countable intersection of α-winning sets is α-winning.(3) An α-winning set in Rn with a countable number of points removed is α-winning.(4) The image of an α-winning set under a bilipschitz map is α′-winning, where α′ dependsonly on α and the map.Now fix a k ∈ {0, 1, · · · , d−1} and restrict 0 < β < 1/3. Bob initially chooses x1 ∈ Rd and r1 > 0and forms a closed ball B1 = B(x1, r1). At each stage of the game, after Bob chooses xi ∈ Rdand ri > 0, Alice chooses an affine subspace L of dimension k and removes an εi-neighborhoodAi = L(εi) from Bi := B(xi, ri) for some 0 < εi ≤ βri. Then Bob chooses xi+1 and ri+1 ≥ βri suchthatBi+1 := B(xi+1, ri+1) ⊂ Bi\Ai.NONDENSE ORBITS FOR ANOSOV DIFFEOMORPHISMS OF THE 2-TORUS 3A set S is said to be k-dimensionally β-absolute winning if Alice has a strategy guaranteeing that∩iBi intersects S. A set S is k-dimensionally absolute winning if it is k-dimensionally β-absolutewinning for every 0 < β < 1/3. We call (d − 1)-dimensionally absolute winning sets hyperplaneabsolute winning (HAW) sets. These sets have three important properties for us [4, Proposition 2.3]:Lemma 2.2. Properties of k-dimensionally absolute winning sets.(1) HAW (and thus k-dimensional absolute winning for all 0 ≤ k ≤ d − 1) implies α-winningfor all 0 < α < 1/2.(2) The countable intersection of k-dimensionally absolute winning sets is k-dimensionally ab-solute winning.(3) The image of a k-dimensionally absolute winning set under a C1 diffeomorphism of Rd isk-dimensionally absolute winning.3. ProofsIn this section, we prove Theorem 1.1 and its corollary. The two main ingredients are the HAWproperty, already introduced, and the following smooth classification of Anosov diffeomorphisms ofthe 2-torus (see [9, Corollary 20.4.5] for example):Theorem 3.1. Suppose g : T2 → T2 is a C2-Anosov diffeomorphism preserving λ such that hλ(g) =htop(g). Then g is C1 conjugate to a linear automorphism.Proof of Theorem 1.1. The Anosov diffeomorphism g is C1-conjugate to a linear automorphism Tby Theorem 3.1. Let h : T2 → T2 be this conjugacy from the Anosov diffeomorphism to thelinear automorphism. The nondense set ND(T, h(x0)) is HAW by [4, Theorem 2.6]. Consequently,ND(g,x0) is HAW by (3) from Lemma 2.2. This proves the desired result. Proof of Corollary 1.2. The desired result follows from the theorem and (1) and (2) from Lemma 2.2and (1) from Lemma 2.1. 3.1. Remarks. A key ingredient in the proof of the smooth classification is the fact that the stableand unstable foliations for a C2-Anosov diffeomorphism of the 2-torus are C1. The proof of thefact relies on a more general result, namely that a codimension-1 stable manifold for a C2-Anosovdiffeomorphism of a compact manifold forms a C1 foliation [8, Corollary 4]. The same applies tothe unstable manifold, from which we obtain the desired C1 foliations for the 2-torus. This leadsto a characterization of smooth conjugacies for the 2-torus [11, 13, 14] (also see [9, Chapter 20.4.b])and to our smooth classification. (Note that, for a torus of dimension three or greater, either theunstable manifold or the stable manifold will not be codimension-1.)For dimensions three and greater, we lack a smooth classification theorem. In fact, the charac-terization of smooth conjugacies is more complicated, and R. de la Llave showed that two Anosovdiffeomorphisms of Td for d ≥ 4 may be topologically conjugate without being Lipschitz conju-gate [12, Theorem 6.3]. (For the 3-torus, the situation is not settled.) A. Gogolev noted that de laLlave’s counterexample can be generalized so that (in most cases) one can take one of the diffeo-morphisms to be a linear automorphism [7, Theorem B]. Should a smooth classification in higherdimensions be found, it would be more complicated than Theorem 3.1, especially since it mustallow for cases in which the conjugacy is only Ho¨lder. On the other hand, we always have a Ho¨lderclassification theorem for this context: every Anosov diffeomorphism of Td is Ho¨lder conjugate toa linear hyperbolic automorphism of Td [9, Theorems 18.6.1 and 19.1.2]. This Ho¨lder classificationis, however, not strong enough for us because we need the conjugacy to be quasisymmetric orbilipschitz (see [16] and (4) from Lemma 2.1) to carry out the proof in this paper.Another way of approaching this problem for dimensions three and greater that should avoidusing a smooth classification is to employ a technique used earlier [18, 21], namely using a boundeddistortion property (i.e. universal bounds on the ratio of the absolute value of the Jacobian deter-minant for nearby points—see [18, Section 2.1.1] for example). If we assume conformality, bounded4 JIMMY TSENGdistortion follows. However, without this assumption (in dimensions two or greater), bounded dis-tortion may not follow, and without it we may not be able to control the local “shape” of successiveiterates of small balls. Since the variants of the game involve intersections with balls, this lack ofcontrol is an obstacle that needs to be overcome if this approach were to succeed for more generalnonlinear maps.Also, we note that for linear maps, one can replace bounded distortion with linear properties ofthe map. In particular, one can decompose Rd into a direct sum of subspaces appropriate for thelinear action and, using the HAW property, one is allowed to remove “bad” affine hyperplanes [4].Linearity seems important to this step and may not possible with a nonlinear system.Finally, we mention a theorem closely related to our results. For pairs of toral mappings, onecan consider the set of points with orbits that are nondense under one of the pair and dense underthe other, a problem first considered in [2]. The nonlinear case for this setup of simultaneous denseand nondense orbits is also interesting, and, for generic pairs of C2-Anosov diffeomorphisms of T2,it is known that the desired set is uncountable and dense [19, Theorem 1.3].References[1] A. G. Abercrombie and R. Nair, An exceptional set in the ergodic theory of expanding maps on manifolds,Monatsh. Math. 148 (2006), 1–17.[2] V. Bergelson, M. Einsiedler, and J. Tseng, Simultaneous dense and nondense orbits for commuting maps, IsraelJ. Math. 210 (2015), 23–45.[3] R. Broderick, L. Fishman, and D. Kleinbock, Schmidt’s game, fractals, and orbits of toral endomorphisms,Ergodic Theory Dynam. Systems 31 (2011), 1095–1107.[4] R. Broderick, L. Fishman, D. Kleinbock, A. Reich, and B. Weiss, The set of badly approximable vectors is stronglyC1 incompressible. Math. 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Tseng, Schmidt games and Markov partitions, Nonlinearity 22 (2009), 525-543.[19] J. Tseng, Simultaneous dense and nondense orbits for toral diffeomorphisms, Ergodic Theory and DynamicalSystems, to appear. http://arxiv.org/abs/1406.1970[20] M. Urban´ski, The Hausdorff dimension of the set of points with nondense orbit under a hyperbolic dynamicalsystem, Nonlinearity 4 (1991), 385–397.[21] W. Wu, Schmidt games and non-dense forward orbits of certain partially hyperbolic systems, Ergodic Theoryand Dynamical Systems, to appear. http://dx.doi.org/10.1017/etds.2014.136NONDENSE ORBITS FOR ANOSOV DIFFEOMORPHISMS OF THE 2-TORUS 5School of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW UKE-mail address: j.tseng@bristol.ac.uk jimmytseng01@gmail.com

Nondense orbits for Anosov Diffeomorphisms of the 2-Torus

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Nondense orbits for Anosov diffeomorphisms of the $2$ -torus

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Nondense orbits for Anosov diffeomorphisms of the 222-torus

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Nondense orbits for Anosov diffeomorphisms of the $2$ -torus